A266521 - OEIS

%I #13 Sep 30 2018 07:12:21

%S 1,2,1,15,28,18,4,683,1278,933,316,42,62038,117440,92680,38240,8272,

%T 752,9342629,17880090,14855385,6881640,1880340,288048,19360,

%U 2100483216,4054752672,3490688496,1743156480,547098240,108228192,12523584,654912,658746323647,1279910119670,1130161189549,594323331364,204256939502,47125635760,7147508032,652959872,27546736,274730459045232,536368375356928,482514140459520,263340552849920,96404466197760,24628940050176,4404380994048,533057051648,39701769216,1388207872

%N E.g.f.: Log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! )  =  Sum_{n>=1} Sum_{k=0..n+1} T(n,k) * x^n*y^k/n!, as a triangle of coefficients T(n,k) read by rows.

%C Row sums form A266520, coefficients in Log( Sum_{n>=0} (n+1)^(2*n) * x^n/n! ).

%C Column 0 forms A266519, coefficients in log( Sum_{n>=0} n^(2*n) * x^n/n! ).

%C Rightmost border is A266526.

%H Paul D. Hanna, <a href="/A266521/b266521.txt">Table of n, a(n) for n = 1..1125 as a flattened triangle read by rows 1..45.</a>

%e E.g.f.: A(x,y) = x * (1 + 2*y + y^2) +

%e x^2/2! * (15 + 28*y + 18*y^2 + 4*y^3) +

%e x^3/3! * (683 + 1278*y + 933*y^2 + 316*y^3 + 42*y^4) +

%e x^4/4! * (62038 + 117440*y + 92680*y^2 + 38240*y^3 + 8272*y^4 + 752*y^5) +

%e x^5/5! * (9342629 + 17880090*y + 14855385*y^2 + 6881640*y^3 + 1880340*y^4 + 288048*y^5 + 19360*y^6) +

%e x^6/6! * (2100483216 + 4054752672*y + 3490688496*y^2 + 1743156480*y^3 + 547098240*y^4 + 108228192*y^5 + 12523584*y^6 + 654912*y^7) +...

%e where

%e exp(A(x,y)) = 1 + (1 + y)*x + (2 + y)^4*x^2/2! + (3 + y)^6*x^3/3! + (4 + y)^8*x^4/4! + (5 + y)^10*x^5/5! + (6 + y)^12*x^6/6! +...

%e This triangle begins:

%e 1, 2, 1;

%e 15, 28, 18, 4;

%e 683, 1278, 933, 316, 42;

%e 62038, 117440, 92680, 38240, 8272, 752;

%e 9342629, 17880090, 14855385, 6881640, 1880340, 288048, 19360;

%e 2100483216, 4054752672, 3490688496, 1743156480, 547098240, 108228192, 12523584, 654912;

%e 658746323647, 1279910119670, 1130161189549, 594323331364, 204256939502, 47125635760, 7147508032, 652959872, 27546736;

%e 274730459045232, 536368375356928, 482514140459520, 263340552849920, 96404466197760, 24628940050176, 4404380994048, 533057051648, 39701769216, 1388207872;

%e 147034646085347145, 288100398039817266, 262835789583073329, 147457696629622032, 56514667400140392, 15510808994500512, 3097157140510272, 445604738641920, 44324678623680, 2758053332736, 81621893376; ...

%o (PARI) {T(n,k) = n! * polcoeff( polcoeff( log( sum(m=0,n+1, (m + y)^(2*m) *x^m/m! ) +x*O(x^n) ),n,x), k,y)}

%o for(n=1,10, for(k=0,n+1, print1(T(n,k),", "));print(""))

%Y Cf. A266519, A266520, A266526.

%K nonn,tabf

%O 1,2

%A _Paul D. Hanna_, Jan 01 2016