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

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

Offset: 1

Author

Donald Mintz (djmintz(AT)home.com)

Keywords

• nonn
• nice

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 in A002144 (Pythagorean primes). - 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

Also number of non-descending sequences with length and last number are less or equal to n, and also the number of integer partitions (of any positive integer) with length and largest part are less or equal to n. - Zlatko Damijanic, Dec 06 2024

Examples

			G.f. = x + 5*x^2 + 19*x^3 + 69*x^4 + 251*x^5 + 923*x^6 + 3431*x^7 + ...
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..500
• Narcisse G. Bell Bogmis, Guy R. Biyogmam, Hesam Safa, and Calvin Tcheka, Upper bounds on the dimension of the Schur Lie-multiplier of Lie-nilpotent Leibniz n-algebras, arXiv:2403.14884 [math.RA], 2024. See p. 7.
• Joseph D. Horton and Andrew Kurn, Counting sequences with complete increasing subsequences, Congress Numerantium, 33 (1981), 75-80. MR 681905
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
• Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
• 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.

Crossrefs

Cf. A000984, A001700, A002144, A051924, A091908, A144660, A322938.

Column k=2 of A047909.

Central column of triangle A014473.

Right-hand column 2 of triangle A102541.

Programs

Magma

[(n+1)*Catalan(n)-1: n in [1..40]]; // G. C. Greubel, Apr 07 2024

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 *)

PARI

a(n)=binomial(2*n,n)-1 \\ Charles R Greathouse IV, Jun 26 2013

Python

from math import comb
def a(n): return comb(2*n, n) - 1
print([a(n) for n in range(1, 27)]) # Michael S. Branicky, Jul 11 2023

Sage

def a(n) : return binomial(2*n,n) - 1
[a(n) for n in (1..26)] # Peter Luschny, Apr 21 2012

Formula

a(n) = A000984(n) - 1.

a(n) = 2*A001700(n-1) - 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_{j=1..n} Sum_{i=1..n} (2n-i-j)!/((n-i)!*(n-j)!). - 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

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

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)

a(n) = 3*n*Sum_{k=1..n} (-1)^(k+1)/(2*n+k)*binomial(2*n+k,n-k). - Vladimir Kruchinin, Jul 29 2025

a(n) = n * binomial(2*n, n) * Sum_{k = 1..n} 1/(k*binomial(n+k, k)). - Peter Bala, Aug 05 2025