A065942 Central column of triangle A065941.

1, 1, 3, 4, 15, 21, 84, 120, 495, 715, 3003, 4368, 18564, 27132, 116280, 170544, 735471, 1081575, 4686825, 6906900, 30045015, 44352165, 193536720, 286097760, 1251677700, 1852482996, 8122425444, 12033222880, 52860229080, 78378960360

Offset: 0

Author

Len Smiley, Nov 29 2001

Keywords

• nonn
• easy

Comments

When viewed as (1,1), (3,4), (15,21), ... this represents a shallow staircase on Pascal's triangle, arranged as a square array. - Paul Barry, Mar 11 2003

Also central column of triangle A011973 (taking rows with odd number of terms only). - John Molokach, Jul 08 2013

Interleaving of A005809 and A045721. - Bruce J. Nicholson, Apr 24 2018

Examples

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 15*x^4 + 21*x^5 + 84*x^6 + 120*x^7 + ... - _Michael Somos_, Jun 23 2018
		

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2416
• Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271; Corrections.

Crossrefs

Cf. A065941 (complete triangle), A047749.

Cf. A005809, A045721.

Programs

GAP

List([0..40],n->Binomial(n+Int(n/2),n)); # Muniru A Asiru, Apr 28 2018

Mathematica

Array[Binomial[# + Floor[#/2], #] &, 30, 0] (* Michael De Vlieger, Apr 27 2018 *)

PARI

a(n) = binomial(n+n\2, n); \\ Altug Alkan, Apr 24 2018

Formula

a(n) = binomial(2n-floor((n+1)/2), floor(n/2)).

a(n+1) = Sum_{k=0..ceiling(n/2)} binomial(n+k, k). - Benoit Cloitre, Mar 06 2004

a(n) = binomial(n+floor(n/2), n). - Paul Barry, May 18 2004

a(n) = Sum_{k=0..floor(n/2)} binomial(n-1+k, k). - Paul Barry, Jul 06 2004

a(2n-1) = binomial(3n-3,n-1); a(2n) = binomial(3n-2,n-1). - John Molokach, Jul 08 2013

G.f.: A(x) = x*(d/dx)[log(S(x)-1)] = x*[(d/dx) S(x)]/[S(x)-1], where S(x) is the g.f. of A047749. - Vladimir Kruchinin, Jun 12 2014.

Conjecture: 8*n*(n-1)*a(n) -36*(n-1)*(n-3)*a(n-1) +6*(-9*n^2+18*n-14)*a(n-2) +27*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Jun 13 2014

0 = a(n)*([phone]*a(n+2) [phone]*a(n+3) -77071527*a(n+4) -134472793*a(n+5)) +a(n+1)*([phone]*a(n+2) -1650969*a(n+3) -9342280*a(n+4) -1729448*a(n+5)) +a(n+2)*(-19089675*a(n+2) -61394833*a(n+3) +6470716*a(n+4) [phone]*a(n+5)) +a(n+3)*(-1291668*a(n+3) +553572*a(n+4) +246032*a(n+5)) for all n in Z. - Michael Somos, Jun 23 2018