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 A030662 Number of combinations of n things from 1 to n at a time, with repeats allowed. 22
 1, 5, 19, 69, 251, 923, 3431, 12869, 48619, 184755, 705431, 2704155, 10400599, 40116599, 155117519, 601080389, 2333606219, 9075135299, 35345263799, 137846528819, 538257874439, 2104098963719, 8233430727599, 32247603683099, 126410606437751, 495918532948103 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Add terms of an increasingly bigger diamond-shaped part of Pascal's triangle: .......................... 1 ............ 1 .......... 1 1 .. 1 ...... 1 1 ........ 1 2 1 . 1 1 =5 . 1 2 1 =19 .. 1 3 3 1 =69 .. 2 ...... 3 3 ........ 4 6 4 ............ 6 ......... 10 10 .......................... 20 -  Ralf Stephan, May 17 2004 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 Also, number of square submatrices of a square matrix. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008 Partial sums of A051924. - J. M. Bergot, Jun 22 2013 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 LINKS T. D. Noe, Table of n, a(n) for n = 1..500 J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congress Numerantium, 33 (1981), 75-80. MR 681905 M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013 M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5 Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014. FORMULA a(n) = A000984(n) - 1. a(n) = 2*(2*n-1)!/(n!*(n-1)!)-1. a(n) = Sum_{k=1..n} binomial(n, k)^2. - Benoit Cloitre, Aug 20 2002 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 a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1} binomial(i+j, i). - N. J. A. Sloane, Jan 31 2009 Also for n>1: a(n)=(2*n)!/(n!)^2-1 - Hugo Pfoertner, Feb 10 2004 a(n) = Sum[Sum[(2n-i-j)!/(n-i)!/(n-j)!,{i,1,n}],{j,1,n}]. - Alexander Adamchuk, Jul 04 2006 a(n) = A115112(n) + 1. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008 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 Conjecture: 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 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 From Ilya Gutkovskiy, Jan 25 2017: (Start) O.g.f.: (1 - x - sqrt(1 - 4*x))/((1 - x)*sqrt(1 - 4*x)). E.g.f.: exp(x)*(exp(x)*BesselI(0,2*x) - 1). (End) EXAMPLE G.f. = x + 5*x^2 + 19*x^3 + 69*x^4 + 251*x^5 + 923*x^6 + 3431*x^7 + ... MAPLE seq(sum((binomial(n, m))^2, m=1..n), n=1..23); # Zerinvary Lajos, Jun 19 2008 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 MATHEMATICA Table[Sum[Sum[(2n-i-j)!/(n-i)!/(n-j)!, {i, 1, n}], {j, 1, n}], {n, 1, 20}] (* Alexander Adamchuk, Jul 04 2006 *) 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 *) PROG (Sage) def a(n) : return binomial(2*n, n) - 1 [a(n) for n in (1..26)] # Peter Luschny, Apr 21 2012 (PARI) a(n)=binomial(2*n, n)-1 \\ Charles R Greathouse IV, Jun 26 2013 CROSSREFS 2*A001700 - 1. Column k=2 of A047909. Cf. A091908, A144660, A002144. Central column of triangle A014473. Right-hand column 2 of triangle A102541. Sequence in context: A240525 A264200 A055991 * A149758 A026590 A243413 Adjacent sequences:  A030659 A030660 A030661 * A030663 A030664 A030665 KEYWORD nonn,nice AUTHOR Donald Mintz (djmintz(AT)home.com) STATUS approved

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Last modified November 19 04:22 EST 2018. Contains 317333 sequences. (Running on oeis4.)