0,5

Cf. the triangle of odd-numbered terms in rows of Pascal's triangle (A034867).

p[n](k), n=0,1,...

k=0: 0, 1, 0, 1, 0, 1, ........... A000035, (A059841)

k=1: 0, 1, 2, 4, 8, 16, .......... A131577, (A000079)

k=2: 0, 1, 4, 13, 40, 121, ....... A003462

k=3: 0, 1, 6, 28, 120, 496, ...... A006516

k=4: 0, 1, 8, 49, 272, 1441, ..... A005059

k=5: 0, 1, 10, 76, 520, 3376, .... A081199, (A016149)

k=6: 0, 1, 12, 109, 888, 6841, ... A081200, (A016161)

k=7: 0, 1, 14, 148, 1400, 12496,.. A081201, (A016170)

k=8: 0, 1, 16, 193, 2080, 21121,.. A081202, (A016178)

k=9: 0, 1, 18, 244, 2952, 33616,.. A081203, (A016186)

k=10:0, 1, 20, 301, 4040, 51001,.. ......., (A016190)

p[n](k), k=0,1,...

p[0]: 0, 0, 0, 0, 0, 0, .......... A000004

p[1]: 1, 1, 1, 1, 1, 1, .......... A000012

p[2]: 0, 2, 4, 6, 8, 10, ......... A005843

p[3]: 1, 4, 13, 28, 49, 76, ...... A056107

p[4]: 0, 8, 40, 120, 272, 520, ... A105374

p[5]: 1, 16, 121, 496, 1441, 3376, ...

p[6]: 0, 32, 364, 2016, 7448, 21280, ..

p_n(x) = sum_{k=0..n} (k mod 2)*binomial(n,k)*x^(n-k)

E.g.f. exp(x*t)/csch(t) = 0*(t^0/0!)+1*(t^1/1!)+(2*x)*(t^2/2!)+(3*x^2+1)*(t^3/3!) + ...

The 'co'-polynomials with generating function exp(x*t)*sech(t) are the Swiss-Knife polynomials (A153641).

0

1, 0

0, 2, 0

1, 0, 3, 0

0, 4, 0, 4, 0

1, 0, 10, 0, 5, 0

0, 6, 0, 20, 0, 6, 0

1, 0, 21, 0, 35, 0, 7, 0

p[0](x) = 0;

p[1](x) = 1

p[2](x) = 2*x

p[3](x) = 3*x^2 + 1

p[4](x) = 4*x^3 + 4*x

p[5](x) = 5*x^4 + 10*x^2 + 1

p[6](x) = 6*x^5 + 20*x^3 + 6*x

p[7](x) = 7*x^6 + 35*x^4 + 21*x^2 + 1

p[8](x) = 8*x^7 + 56*x^5 + 56*x^3 + 8*x

# Polynomials: p_n(x)

p := proc(n, x) local k;

pow := (n, k) -> `if`(n=0 and k=0, 1, n^k);

add((k mod 2)*binomial(n, k)*pow(x, n-k), k=0..n) end;

# Coefficients: a(n)

seq(print(seq(coeff(i!*coeff(series(exp(x*t)/csch(t), t, 16), t, i), x, n), n=0..i)), i=0..8);

Sequence in context: A103775 A093057 A065334 this_sequence A007814 A083280 A060689

Adjacent sequences: A162587 A162588 A162589 this_sequence A162591 A162592 A162593

nonn,tabl

Peter Luschny (peter(AT)luschny.de), Jul 07 2009