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%I #49 Nov 20 2021 08:02:36
%S 1,7,105,2386,71890,2695652,120907185,6312179764,375971507406,
%T 25160695768715,1869031937691061,152603843369288819,
%U 13584174777196666630,1309317592648179024666,135850890740575408906465
%N Sum of the terms of the n-th row of triangle pertaining to A096130.
%C The product of the terms of the n-th row is given by A034841.
%C Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - _Olivier Gérard_, Aug 08 2016
%H Seiichi Manyama, <a href="/A096131/b096131.txt">Table of n, a(n) for n = 1..338</a>
%F a(n) = Sum_{k=1..n} binomial(k*n, n). - _Reinhard Zumkeller_, Jan 09 2005
%F a(n) = (1/n!) * Sum_{j=1..n} Product_{i=n*(j-1)+1..n*j} i. - _Reinhard Zumkeller_, Jan 09 2005 [corrected by _Seiichi Manyama_, Aug 17 2018]
%F a(n) ~ exp(1)/(exp(1)-1) * binomial(n^2,n). - _Vaclav Kotesovec_, Jun 06 2013
%e From _Seiichi Manyama_, Aug 18 2018: (Start)
%e a(1) = (1/1!) * (1) = 1.
%e a(2) = (1/2!) * (1*2 + 3*4) = 7.
%e a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
%e a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)
%p A096130 := proc(n,k) binomial(k*n,n) ; end: A096131 := proc(n) local k; add( A096130(n,k),k=1..n) ; end: for n from 1 to 18 do printf("%d, ",A096131(n)) ; od ; # _R. J. Mathar_, Apr 30 2007
%p seq(add((binomial(n*k,n)), k=0..n), n=1..15); # _Zerinvary Lajos_, Sep 16 2007
%t Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Jun 06 2013 *)
%o (GAP) List(List([1..20],n->List([1..n],k->Binomial(k*n,n))),Sum); # _Muniru A Asiru_, Aug 12 2018
%o (PARI) a(n) = sum(k=1, n, binomial(k*n, n)); \\ _Michel Marcus_, Aug 20 2018
%Y Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Jul 04 2004
%E More terms from _R. J. Mathar_, Apr 30 2007
%E Edited by _N. J. A. Sloane_, Sep 06 2008 at the suggestion of _R. J. Mathar_
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