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A162590 Polynomials with e.g.f. exp(x*t)/csch(t), triangle of coefficients read by rows. 6
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, 0, 8, 0, 56, 0, 56, 0, 8, 0, 1, 0, 36, 0, 126, 0, 84, 0, 9, 0, 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 1, 0, 55, 0, 330, 0, 462, 0, 165, 0, 11, 0, 0, 12, 0, 220, 0, 792, 0, 792, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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, ...

Comment from Peter Bala (Dec 06 2011): "Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of (I-t*M)^-1 (apart from the initial 1) lists the row polynomials for" A196776(n,k), which gives the number of ordered partitions of an n set into k odd-sized blocks. - Peter Luschny, Dec 06 2011

LINKS

Table of n, a(n) for n=0..86.

FORMULA

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).

EXAMPLE

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

MAPLE

# 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);

MATHEMATICA

p[n_, x_] := Sum[Binomial[n, 2*k-1]*x^(n-2*k+1), {k, 0, n+2}]; row[n_] := CoefficientList[p[n, x], x] // Append[#, 0]&; Table[row[n], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Jun 28 2013 *)

CROSSREFS

Sequence in context: A103775 A093057 A065334 * A276424 A191258 A254990

Adjacent sequences:  A162587 A162588 A162589 * A162591 A162592 A162593

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jul 07 2009

STATUS

approved

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Last modified June 21 23:05 EDT 2018. Contains 305646 sequences. (Running on oeis4.)