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Triangular sequence from a Pidduck polynomials expansion: p(t) = (t/(1 - t))*((1 + t)/(1 - t))^x.
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%I #12 May 28 2023 08:46:12

%S 1,2,4,6,12,12,24,64,48,32,120,320,400,160,80,720,2208,2400,1920,480,

%T 192,5040,15456,21952,13440,7840,1344,448,40320,135168,175616,157696,

%U 62720,28672,3584,1024,362880,1216512,1884672,1419264,919296,258048

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

%C 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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PidduckPolynomial.html">Pidduck Polynomial</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pidduck_polynomials">Pidduck polynomials</a>

%F 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))

%e {0},

%e {1},

%e {2, 4},

%e {6, 12, 12},

%e {24, 64, 48, 32},

%e {120, 320, 400, 160, 80},

%e {720, 2208, 2400, 1920, 480, 192},

%e {5040, 15456, 21952, 13440, 7840, 1344, 448},

%e {40320, 135168, 175616, 157696, 62720, 28672, 3584, 1024},

%e {362880, 1216512, 1884672, 1419264, 919296, 258048, 96768, 9216, 2304}, {3628800, 12971520, 18846720, 18380800, 9192960, 4623360, 967680, 307200, 23040, 5120}

%t p[t_] = (t/(1 - t))*((1 + t)/(1 - t))^x

%t Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}];

%t a = Table[ CoefficientList[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}];

%t Flatten[a]

%K nonn,uned,tabf

%O 1,2

%A _Roger L. Bagula_, Apr 10 2008