A137393 Triangular sequence from a Peters polynomials expansion: l0 = 2; m0 = 2; p(t) = (1 + t)^x/(1 + (1 + t)^l0)^m0.

1, -4, 2, 16, -20, 4, -48, 160, -72, 8, -96, -1120, 944, -224, 16, 3840, 6208, -10880, 4320, -640, 32, -46080, -12672, 115456, -72000, 16960, -1728, 64, 322560, -294912, -1146880, 1121792, -380800, 60032, -4480, 128, 645120, 4663296, 11223040, -17110016, 7933184, -1734656, 197120, -11264, 256

Offset

1,2

Comments

Row sums are: {1, -2, 0, 48, -480, 2880, 0, -322560, 5806080, -58060800, 0}

s(n,x,lambda,mu) = (n!*sum(i=0..n, binomial(x,n-i)*sum(k=0..i, (binomial(mu+k-1,k) * sum(j=0..k, (-1)^j*binomial(k,j)* binomial(j*lambda,i)))/ 2^k)))/2^mu. - Vladimir Kruchinin, Jan 12 2012

Links

Table of n, a(n) for n=1..45.

V. Kruchinin, D. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, arXiv: 1211.0099

Eric Weisstein's World of Mathematics, Peters Polynomial.

Formula

l0 = 2; m0 = 2; p(t) = (1 + t)^x/(1 + (1 + t)^l0)^m0=Sum(s(x,k,l0,m0)*t^k/k!,{k,0,Infinity}]; Out(n,m)=2^(n+2)*n!*Coefficient(s(x,n,2,2))

T(n,r) = 2^n*sum(j=0..n, C(n,j)*((sum(i=0..j, (-1)^i*2^i*sum(m=i..j,((sum(k=m..j, stirling1(j,k)*2^k*stirling2(k,m))) *stirling1(m,i))/2^m))) *stirling1(n-j,r))). - Vladimir Kruchinin, Jan 12 2012

Example

{1},
{-4, 2},
{16, -20, 4},
{-48, 160, -72, 8},
{-96, -1120, 944, -224, 16},
{3840, 6208, -10880, 4320, -640, 32},
{-46080, -12672, 115456, -72000, 16960, -1728, 64},
{322560, -294912, -1146880, 1121792, -380800, 60032, -4480, 128},
{645120, 4663296, 11223040, -17110016, 7933184, -1734656, 197120, -11264, 256},
{-69672960, -1363968, -133447680, 268337152, -161344512, 45943296, -7096320, 611328, -27648, 512}, {1393459200, -1720442880, 2586968064, -4625121280, 3334430720, -1175516160, 231232512, -26757120, 1812480, -66560, 1024}

Mathematica

Clear[p, l0, m0] l0 = 2; m0 = 2; p[t_] = (1 + t)^x/(1 + (1 + t)^l0)^m0 Table[ ExpandAll[2^(n + 2)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[2^(n + 2)*n! *SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

Prog

(Maxima) T(n, r):=2^n*sum(binomial(n, j)*((sum((-1)^i*2^i* sum(((sum( stirling1(j, k)*2^k*stirling2(k, m), k, m, j)) *stirling1(m, i))/2^m, m, i, j), i, 0, j))*stirling1(n-j, r)), j, 0, n); /* Vladimir Kruchinin, Jan 12 2012 */

Crossrefs

Sequence in context: A084623 A264195 A182872 * A356569 A122749 A189741

Adjacent sequences: A137390 A137391 A137392 * A137394 A137395 A137396

Keyword

uned,tabl,sign

Author

Roger L. Bagula, Apr 10 2008

Status

approved