A137394 Triangular sequence from a Pidduck polynomials expansion: p(t) = (t/(1 - t))*((1 + t)/(1 - t))^x.

1, 2, 4, 6, 12, 12, 24, 64, 48, 32, 120, 320, 400, 160, 80, 720, 2208, 2400, 1920, 480, 192, 5040, 15456, 21952, 13440, 7840, 1344, 448, 40320, 135168, 175616, 157696, 62720, 28672, 3584, 1024, 362880, 1216512, 1884672, 1419264, 919296, 258048

Offset

1,2

Comments

The Wikipedia article has the correct definition of Pidduck polynomials. It uses (1/(1 - t))*((1 + t)/(1 - t))^x. The MathWorld article equation (3) has a typo which is copied here. Note that the values P_0(x) = 1, P_1(x) = 2x+1, P_2(x) = 4x^2 + 4x + 2, etc. given in MathWorld are correct though. - Michael Somos, May 27 2023

Links

Table of n, a(n) for n=1..42.

Eric Weisstein's World of Mathematics, Pidduck Polynomial.

Wikipedia, Pidduck polynomials

Formula

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

Example

{0},
{1},
{2, 4},
{6, 12, 12},
{24, 64, 48, 32},
{120, 320, 400, 160, 80},
{720, 2208, 2400, 1920, 480, 192},
{5040, 15456, 21952, 13440, 7840, 1344, 448},
{40320, 135168, 175616, 157696, 62720, 28672, 3584, 1024},
{362880, 1216512, 1884672, 1419264, 919296, 258048, 96768, 9216, 2304}, {3628800, 12971520, 18846720, 18380800, 9192960, 4623360, 967680, 307200, 23040, 5120}

Mathematica

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

Crossrefs

Sequence in context: A076868 A056793 A137387 * A062856 A056371 A271822

Adjacent sequences: A137391 A137392 A137393 * A137395 A137396 A137397

Keyword

nonn,uned,tabf

Author

Roger L. Bagula, Apr 10 2008

Status

approved