A316590 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.

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

Offset

1,3

Comments

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

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

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

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

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1035 of terms in rows 1..45 of the flattened form of this triangle.

Example

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) + ...
such that
Sum_{m>=0} (x^m + y + 1/x^m)^m = A(x,y) + A(1/x,y) + (infinity)*x^0.
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:
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, 280, 6, 560, 0, 168, 0, 8;
127, 0, 1260, 0, 1260, 0, 252, 0, 9;
0, 1280, 0, 4210, 0, 2520, 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;
24310, 0, 875160, 0, 4084080, 0, 5717712, 0, 3063060, 0, 680680, 0, 61880, 0, 2040, 0, 17;
0, 438114, 0, 5252240, 0, 14703192, 0, 14702724, 0, 6126120, 0, 1113840, 0, 85680, 0, 2448, 0, 18;
92378, 0, 4157010, 0, 24942060, 0, 46558512, 0, 34918884, 0, 11639628, 0, 1763580, 0, 116280, 0, 2907, 0, 19;
210, 1847565, 2520, 27713400, 3150, 99768240, 840, 133024320, 45, 77597520, 0, 21162960, 0, 2713200, 0, 155040, 0, 3420, 0, 20;
...
Column 0 of this triangle equals A304638.
Row sums of this triangle yields A316591.

Prog

(PARI) {T(n, k) = polcoeff( polcoeff( sum(m=1, n, (x^-m + y + x^m)^m +x*O(x^n)), n, x), k, y)}
for(n=1, 20, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

Crossrefs

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

Sequence in context: A334291 A051910 A137998 * A080593 A319148 A193682

Adjacent sequences: A316587 A316588 A316589 * A316591 A316592 A316593

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 08 2018

Status

approved