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%I #103 Apr 08 2024 09:14:24
%S 1,5,19,69,251,923,3431,12869,48619,184755,705431,2704155,10400599,
%T 40116599,155117519,601080389,2333606219,9075135299,35345263799,
%U 137846528819,538257874439,2104098963719,8233430727599,32247603683099,126410606437751,495918532948103
%N Number of combinations of n things from 1 to n at a time, with repeats allowed.
%C Add terms of an increasingly bigger diamond-shaped part of Pascal's triangle:
%C .......................... 1
%C ............ 1 .......... 1 1
%C .. 1 ...... 1 1 ........ 1 2 1
%C . 1 1 =5 . 1 2 1 =19 .. 1 3 3 1 =69
%C .. 2 ...... 3 3 ........ 4 6 4
%C ............ 6 ......... 10 10
%C .......................... 20
%C - _Ralf Stephan_, May 17 2004
%C The prime p divides a((p-1)/2) for p in A002144 (Pythagorean primes). - _Alexander Adamchuk_, Jul 04 2006
%C Also, number of square submatrices of a square matrix. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008
%C Partial sums of A051924. - _J. M. Bergot_, Jun 22 2013
%C Number of partitions with Ferrers diagrams that fit in an n X n box (excluding the empty partition of 0). - _Michael Somos_, Jun 02 2014
%H T. D. Noe, <a href="/A030662/b030662.txt">Table of n, a(n) for n = 1..500</a>
%H Narcisse G. Bell Bogmis, Guy R. Biyogmam, Hesam Safa, and Calvin Tcheka, <a href="https://arxiv.org/abs/2403.14884">Upper bounds on the dimension of the Schur Lie-multiplier of Lie-nilpotent Leibniz n-algebras</a>, arXiv:2403.14884 [math.RA], 2024. See p. 7.
%H Joseph D. Horton and Andrew Kurn, Counting sequences with complete increasing subsequences, Congress Numerantium, 33 (1981), 75-80. <a href="http://www.ams.org/mathscinet-getitem?mr=681905">MR 681905</a>
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H Raimundas Vidunas, <a href="https://doi.org/10.1007/s00026-017-0344-2">Counting derangements and Nash equilibria</a>, Ann. Comb. 21, No. 1, 131-152 (2017).
%H Jianqiang Zhao, <a href="http://arxiv.org/abs/1412.8044">Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras</a>, arXiv preprint arXiv:1412.8044 [math.NT], 2014.
%F a(n) = A000984(n) - 1.
%F a(n) = 2*A001700(n-1) - 1.
%F a(n) = 2*(2*n-1)!/(n!*(n-1)!)-1.
%F a(n) = Sum_{k=1..n} binomial(n, k)^2. - _Benoit Cloitre_, Aug 20 2002
%F a(n) = Sum_{j=0..n} Sum_{i=j..n+j} binomial(i, j). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 23 2003
%F a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1} binomial(i+j, i). - _N. J. A. Sloane_, Jan 31 2009
%F Also for n>1: a(n)=(2*n)!/(n!)^2-1. - _Hugo Pfoertner_, Feb 10 2004
%F a(n) = Sum_{j=1..n} Sum_{i=1..n} (2n-i-j)!/((n-i)!*(n-j)!). - _Alexander Adamchuk_, Jul 04 2006
%F a(n) = A115112(n) + 1. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008
%F G.f.: Q(0)*(1-4*x)/x - 1/x/(1-x), where Q(k)= 1 + 4*(2*k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 11 2013
%F D-finite with recurrence: n*a(n) +2*(-3*n+2)*a(n-1) +(9*n-14)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jun 25 2013
%F 0 = a(n)*(+16*a(n+1) - 70*a(n+2) + 68*a(n+3) - 14*a(n+4)) + a(n+1)*(-2*a(n+1) + 61*a(n+2) - 96*a(n+3) + 23*a(n+4)) + a(n+2)*(-6*a(n+2) + 31*a(n+3) - 10*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - _Michael Somos_, Jun 02 2014
%F From _Ilya Gutkovskiy_, Jan 25 2017: (Start)
%F O.g.f.: (1 - x - sqrt(1 - 4*x))/((1 - x)*sqrt(1 - 4*x)).
%F E.g.f.: exp(x)*(exp(x)*BesselI(0,2*x) - 1). (End)
%e G.f. = x + 5*x^2 + 19*x^3 + 69*x^4 + 251*x^5 + 923*x^6 + 3431*x^7 + ...
%p seq(sum((binomial(n,m))^2,m=1..n),n=1..23); # _Zerinvary Lajos_, Jun 19 2008
%p f:=n->add( add( binomial(i+j,i), i=0..n),j=0..n); [seq(f(n),n=0..12)]; # _N. J. A. Sloane_, Jan 31 2009
%t Table[Sum[Sum[(2n-i-j)!/(n-i)!/(n-j)!,{i,1,n}],{j,1,n}],{n,1,20}] (* _Alexander Adamchuk_, Jul 04 2006 *)
%t a[n_] := 2*(2*n-1)!/(n*(n-1)!^2)-1; Table[a[n], {n, 1, 26}] (* _Jean-François Alcover_, Oct 11 2012, from first formula *)
%o (Sage)
%o def a(n) : return binomial(2*n,n) - 1
%o [a(n) for n in (1..26)] # _Peter Luschny_, Apr 21 2012
%o (PARI) a(n)=binomial(2*n,n)-1 \\ _Charles R Greathouse IV_, Jun 26 2013
%o (Python)
%o from math import comb
%o def a(n): return comb(2*n, n) - 1
%o print([a(n) for n in range(1, 27)]) # _Michael S. Branicky_, Jul 11 2023
%o (Magma) [(n+1)*Catalan(n)-1: n in [1..40]]; // _G. C. Greubel_, Apr 07 2024
%Y Cf. A000984, A001700, A002144, A051924, A091908, A144660.
%Y Column k=2 of A047909.
%Y Central column of triangle A014473.
%Y Right-hand column 2 of triangle A102541.
%K nonn,nice
%O 1,2
%A Donald Mintz (djmintz(AT)home.com)
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