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 A171685 Triangle T(n,k) which contains 16*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(15 + exp(8*t)) in row n, column k. 0

%I

%S 1,-1,2,-7,-2,2,-83,-42,-6,4,-266,-332,-84,-8,4,5666,-2660,-1660,-280,

%T -20,8,146762,33996,-7980,-3320,-420,-24,8,3415978,2054668,237972,

%U -37240,-11620,-1176,-56,16,7599256,27327824,8218672,634592,-74480

%N Triangle T(n,k) which contains 16*n!*2^floor((n+1)/2) times the coefficient [t^n x^k] exp(t*x)/(15 + exp(8*t)) in row n, column k.

%C The bivariate taylor expansion of exp(t*x)/(15+exp(8*t)) is 1/16 + (x/16-1/32)*t +(-7/64+x^2/32 -x/32)*t^2+ (-83/384+x^3/96-7*x/64-x^2/64)*t^3+...

%C Row n contains the coefficients of the polynomial in front of t^n, multiplied by 16*floor[(n+1)/2]*n!.

%C Row sums are: 1, 1, -7, -127, -686, 1054, 169022, 5658542, 43685656, -1052651384, -55785840712,....

%e The triangle starts in row n=0 with columns 0<=k <=n as

%e 1;

%e -1, 2;

%e -7, -2, 2;

%e -83, -42, -6, 4;

%e -266, -332, -84, -8, 4;

%e 5666, -2660, -1660, -280, -20, 8;

%e 146762, 33996, -7980, -3320, -420, -24, 8;

%e 3415978, 2054668, 237972, -37240, -11620, -1176, -56, 16;

%e 7599256, 27327824, 8218672, 634592, -74480, -18592, -1568, -64, 16;

%e -1487228056, 136786608, 245950416, 49312032, 2855664, -268128, -55776, -4032, -144, 32;

%e -42545787592, -14872280560, 683933040, 819834720, 123280080, 5711328, -446880, -79680, -5040, -160, 32;

%t Clear[p, g, m, a];

%t m = 3;

%t p[t_] = 2^(m + 1)*Exp[t*x]/(-1 + 2^(m + 1) + Exp[2^m*t])

%t Table[ FullSimplify[ExpandAll[2^ Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]

%t a = Table[CoefficientList[FullSimplify[ExpandAll[2^Floor[(n + 1)/2]*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]

%t Flatten[a]

%Y Cf. A000364, A171684.

%K sign,tabl

%O 0,3

%A _Roger L. Bagula_, Dec 15 2009

%E Number of variables in use reduced from 4 to 2, keyword:tabl added - The Assoc. Eds. of the OEIS, Oct 20 2010

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Last modified January 27 21:49 EST 2023. Contains 359849 sequences. (Running on oeis4.)