A299427 Square table where T(n,k) = binomial(n*(n+k), k) * n/(n+k), for n>=1, k>=0, as read by antidiagonals.

1, 1, 1, 4, 1, 1, 9, 14, 1, 1, 16, 63, 48, 1, 1, 25, 184, 408, 165, 1, 1, 36, 425, 1872, 2565, 572, 1, 1, 49, 846, 6175, 17980, 15939, 2002, 1, 1, 64, 1519, 16536, 82775, 167552, 98670, 7072, 1, 1, 81, 2528, 38318, 292581, 1059380, 1535352, 610740, 25194, 1, 1, 100, 3969, 79808, 861175, 4874688, 13177125, 13934752, 3786588, 90440, 1

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1079 of rows 1..45 as a flattened square table read by antidiagonals.

Formula

G.f. for row n: R(x,n)^(n^2) = Sum_{k>=0} C(n*(n+k), k) * n/(n+k) * x^k, where R(x,n) = 1 + x*R(x,n)^n.

Example

This table begins:
n=1: [1,  1,    1,      1,       1,         1,          1, ...];
n=2: [1,  4,   14,     48,     165,       572,       2002, ...];
n=3: [1,  9,   63,    408,    2565,     15939,      98670, ...];
n=4: [1, 16,  184,   1872,   17980,    167552,    1535352, ...];
n=5: [1, 25,  425,   6175,   82775,   1059380,   13177125, ...];
n=6: [1, 36,  846,  16536,  292581,   4874688,   78119454, ...];
n=7: [1, 49, 1519,  38318,  861175,  18008676,  358919022, ...];
n=8: [1, 64, 2528,  79808, 2214640,  56592320, 1367090208, ...];
n=9: [1, 81, 3969, 153117, 5132565, 157000275, 4507103601, ...];
...
Row generating functions R(x,n)^(n^2) begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
R(x,2)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + 572*x^5  + ...
R(x,3)^9 = 1 + 9*x + 63*x^2 + 408*x^3 + 2565*x^4 + 15939*x^5 + ...
R(x,4)^16 = 1 + 16*x + 184*x^2 + 1872*x^3 + 17980*x^4 + 167552*x^5 + ...
R(x,5)^25 = 1 + 25*x + 425*x^2 + 6175*x^3 + 82775*x^4 + 1059380*x^5 + ...
R(x,6)^36 = 1 + 36*x + 846*x^2 + 16536*x^3 + 292581*x^4 + 4874688*x^5 + ...
...
Related series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
where R(x,n)^m = Sum_{k>=0} C(m + n*k, k) * m/(m + n*k) * x^k.
...

Prog

(PARI) {T(n, k) = binomial(n*(n+k), k) * n/(n+k) }
/* Print as a square table of first 9 rows */
for(n=1, 9, print1("n="n": [", ); for(k=0, 8, print1(T(n, k), ", ")); print1("...]; "); print(""))
/* Print as a Flattened table read by antidiagonals */
for(n=1, 10, for(k=0, n, print1(T(n-k+1, k), ", ")))

Crossrefs

Cf. A299044 (antidiagonal sums), A299428 (diagonal), A299429.

Sequence in context: A168621 A039756 A126065 * A126062 A243608 A219207

Adjacent sequences: A299424 A299425 A299426 * A299428 A299429 A299430

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 19 2018

Status

approved