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A299427 Square table where T(n,k) = binomial(n*(n+k), k) * n/(n+k), for n>=1, k>=0, as read by antidiagonals. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 14 08:44 EDT 2021. Contains 343879 sequences. (Running on oeis4.)