A316590 - OEIS

%I #16 Feb 25 2020 01:24:06

%S 1,0,2,3,0,3,1,12,0,4,10,0,30,0,5,0,63,0,60,0,6,35,0,210,0,105,0,7,4,

%T 280,6,560,0,168,0,8,127,0,1260,0,1260,0,252,0,9,0,1280,0,4210,0,2520,

%U 0,360,0,10,462,0,6930,0,11550,0,4620,0,495,0,11,15,5548,60,27720,15,27720,0,7920,0,660,0,12,1716,0,36036,0,90090,0,60060,0,12870,0,858,0,13,0,24129,0,168308,0,252273,0,120120,0,20020,0,1092,0,14,6440,0,180190,0,630630,0,630630,0,225225,0,30030,0,1365,0,15,57,102960,420,960960,280,2018016,28,1441440,0,400400,0,43680,0,1680,0,16

%N G.f. A(x,y) satisfies: A(x,y) + A(1/x,y) = Sum_{m>=0} (x^m + y + 1/x^m)^m, ignoring the infinite constant term; this is the triangle, read by rows, of coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 1, k = 0..n-1.

%C A304638(n) = T(n,0) for n >= 1, and forms the first column of this triangle.

%C A316591(n) = Sum_{k=0..n-1} T(n,k) for n >= 1, giving the row sums.

%C A316592(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.

%C A316593(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.

%C A316594(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.

%C A316595(n) = Sum_{k=0..n-1} T(n,k) * 5^k for n >= 1.

%H Paul D. Hanna, <a href="/A316590/b316590.txt">Table of n, a(n) for n = 1..1035 of terms in rows 1..45 of the flattened form of this triangle.</a>

%e G.f.: A(x,y) = x + x^2*(2*y) + x^3*(3 + 3*y^2) + x^4*(1 + 12*y + 4*y^3) + x^5*(10 + 30*y^2 + 5*y^4) + x^6*(63*y + 60*y^3 + 6*y^5) + x^7*(35 + 210*y^2 + 105*y^4 + 7*y^6) + x^8*(4 + 280*y + 6*y^2 + 560*y^3 + 168*y^5 + 8*y^7) + x^9*(127 + 1260*y^2 + 1260*y^4 + 252*y^6 + 9*y^8) + x^10*(1280*y + 4210*y^3 + 2520*y^5 + 360*y^7 + 10*y^9) + x^11*(462 + 6930*y^2 + 11550*y^4 + 4620*y^6 + 495*y^8 + 11*y^10) + x^12*(15 + 5548*y + 60*y^2 + 27720*y^3 + 15*y^4 + 27720*y^5 + 7920*y^7 + 660*y^9 + 12*y^11) + ...

%e such that

%e Sum_{m>=0} (x^m + y + 1/x^m)^m = A(x,y) + A(1/x,y) + (infinity)*x^0.

%e This triangle of coefficients T(n,k) of x^n*y^k, n >= 1, k = 0..n-1, in g.f. A(x,y) begins:

%e 1;

%e 0, 2;

%e 3, 0, 3;

%e 1, 12, 0, 4;

%e 10, 0, 30, 0, 5;

%e 0, 63, 0, 60, 0, 6;

%e 35, 0, 210, 0, 105, 0, 7;

%e 4, 280, 6, 560, 0, 168, 0, 8;

%e 127, 0, 1260, 0, 1260, 0, 252, 0, 9;

%e 0, 1280, 0, 4210, 0, 2520, 0, 360, 0, 10;

%e 462, 0, 6930, 0, 11550, 0, 4620, 0, 495, 0, 11;

%e 15, 5548, 60, 27720, 15, 27720, 0, 7920, 0, 660, 0, 12;

%e 1716, 0, 36036, 0, 90090, 0, 60060, 0, 12870, 0, 858, 0, 13;

%e 0, 24129, 0, 168308, 0, 252273, 0, 120120, 0, 20020, 0, 1092, 0, 14;

%e 6440, 0, 180190, 0, 630630, 0, 630630, 0, 225225, 0, 30030, 0, 1365, 0, 15;

%e 57, 102960, 420, 960960, 280, 2018016, 28, 1441440, 0, 400400, 0, 43680, 0, 1680, 0, 16;

%e 24310, 0, 875160, 0, 4084080, 0, 5717712, 0, 3063060, 0, 680680, 0, 61880, 0, 2040, 0, 17;

%e 0, 438114, 0, 5252240, 0, 14703192, 0, 14702724, 0, 6126120, 0, 1113840, 0, 85680, 0, 2448, 0, 18;

%e 92378, 0, 4157010, 0, 24942060, 0, 46558512, 0, 34918884, 0, 11639628, 0, 1763580, 0, 116280, 0, 2907, 0, 19;

%e 210, 1847565, 2520, 27713400, 3150, 99768240, 840, 133024320, 45, 77597520, 0, 21162960, 0, 2713200, 0, 155040, 0, 3420, 0, 20;

%e ...

%e Column 0 of this triangle equals A304638.

%e Row sums of this triangle yields A316591.

%o (PARI) {T(n,k) = polcoeff( polcoeff( sum(m=1,n, (x^-m + y + x^m)^m +x*O(x^n)), n,x), k,y)}

%o for(n=1,20, for(k=0,n-1, print1(T(n,k),", "));print(""))

%Y Cf. A304638 (column 0), A316591 (row sums), A316592, A316593, A316594, A316595.

%K nonn,tabl

%O 1,3

%A _Paul D. Hanna_, Jul 08 2018