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 A030662 Number of combinations of n things from 1 to n at a time, with repeats allowed. 25

%I

%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 = 5,13,17,29,37,41,53,61,73,89,97.. = A002144[n] Pythagorean primes: primes of form 4n+1. - _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 J. D. Horton and A. 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 M. Janjic and B. 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 M. Janjic, B. 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 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*(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[Sum[(2n-i-j)!/(n-i)!/(n-j)!,{i,1,n}],{j,1,n}]. - _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

%Y 2*A001700 - 1.

%Y Column k=2 of A047909.

%Y Cf. A091908, A144660, A002144.

%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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Last modified April 7 13:36 EDT 2020. Contains 333305 sequences. (Running on oeis4.)