cp's OEIS Frontend

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

%I A299427 #14 Feb 19 2018 12:27:57
%S A299427 1,1,1,4,1,1,9,14,1,1,16,63,48,1,1,25,184,408,165,1,1,36,425,1872,
%T A299427 2565,572,1,1,49,846,6175,17980,15939,2002,1,1,64,1519,16536,82775,
%U A299427 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
%N A299427 Square table where T(n,k) = binomial(n*(n+k), k) * n/(n+k), for n>=1, k>=0, as read by antidiagonals.
%H A299427 Paul D. Hanna, <a href="/A299427/b299427.txt">Table of n, a(n) for n = 0..1079 of rows 1..45 as a flattened square table read by antidiagonals.</a>
%F A299427 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.
%e A299427 This table begins:
%e A299427 n=1: [1,  1,    1,      1,       1,         1,          1, ...];
%e A299427 n=2: [1,  4,   14,     48,     165,       572,       2002, ...];
%e A299427 n=3: [1,  9,   63,    408,    2565,     15939,      98670, ...];
%e A299427 n=4: [1, 16,  184,   1872,   17980,    167552,    1535352, ...];
%e A299427 n=5: [1, 25,  425,   6175,   82775,   1059380,   13177125, ...];
%e A299427 n=6: [1, 36,  846,  16536,  292581,   4874688,   78119454, ...];
%e A299427 n=7: [1, 49, 1519,  38318,  861175,  18008676,  358919022, ...];
%e A299427 n=8: [1, 64, 2528,  79808, 2214640,  56592320, 1367090208, ...];
%e A299427 n=9: [1, 81, 3969, 153117, 5132565, 157000275, 4507103601, ...];
%e A299427 ...
%e A299427 Row generating functions R(x,n)^(n^2) begin:
%e A299427 R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
%e A299427 R(x,2)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + 572*x^5  + ...
%e A299427 R(x,3)^9 = 1 + 9*x + 63*x^2 + 408*x^3 + 2565*x^4 + 15939*x^5 + ...
%e A299427 R(x,4)^16 = 1 + 16*x + 184*x^2 + 1872*x^3 + 17980*x^4 + 167552*x^5 + ...
%e A299427 R(x,5)^25 = 1 + 25*x + 425*x^2 + 6175*x^3 + 82775*x^4 + 1059380*x^5 + ...
%e A299427 R(x,6)^36 = 1 + 36*x + 846*x^2 + 16536*x^3 + 292581*x^4 + 4874688*x^5 + ...
%e A299427 ...
%e A299427 Related series R(x,n) = 1 + x*R(x,n)^n begin:
%e A299427 R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
%e A299427 R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
%e A299427 R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
%e A299427 R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
%e A299427 R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
%e A299427 R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
%e A299427 ...
%e A299427 where R(x,n)^m = Sum_{k>=0} C(m + n*k, k) * m/(m + n*k) * x^k.
%e A299427 ...
%o A299427 (PARI) {T(n,k) = binomial(n*(n+k), k) * n/(n+k) }
%o A299427 /* Print as a square table of first 9 rows */
%o A299427 for(n=1,9,print1("n="n": [",); for(k=0,8,print1(T(n,k),", ")); print1("...];");print(""))
%o A299427 /* Print as a Flattened table read by antidiagonals */
%o A299427 for(n=1,10,for(k=0,n,print1(T(n-k+1,k),", ")))
%Y A299427 Cf. A299044 (antidiagonal sums), A299428 (diagonal), A299429.
%K A299427 nonn,tabl
%O A299427 0,4
%A A299427 _Paul D. Hanna_, Feb 19 2018