A167660 Chocolate dove bar numerator: a(n) = (Sum_{k=0..floor(n/2)} k*binomial(n+k,k)*binomial(n,n-2*k)) + (Sum_{k=0..ceiling(n/2)} k*binomial(n+k-1,k-1)*binomial(n,n-2*k+1)).

0, 1, 5, 23, 104, 458, 1987, 8523, 36248, 153134, 643466, 2691926, 11220156, 46620412, 193190831, 798700531, 3295291440, 13571239766, 55801698214, 229113328722, 939486081152, 3847872039340, 15742988692542, 64347264994238

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

D. M. Einstein, C. C. Heckman and T. S. Norfolk, On Sara's Dove Bar Habit

D. M. Einstein, C. C. Heckman, and T. S. Norfolk, On Sara's Dove Bar Habit, American Mathematical Monthly, Nov. 2009, p. 831.

Formula

Recurrence: 2*(n-2)*n*a(n) = (3*n^2 + 9*n - 28)*a(n-1) + 2*(9*n^2 - 33*n + 22)*a(n-2) + 4*(n-1)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 4^n*sqrt(n)/(3*sqrt(Pi)). - Vaclav Kotesovec, Oct 20 2012

Mathematica

a[n_]:= Sum[k*Binomial[n + k, k]*Binomial[n, n - 2*k], {k, 0, Floor[ n/2]}] + Sum[k*Binomial[n + k - 1, k - 1]* Binomial[n, n - 2*k + 1], {k, 0, Floor[(n + 1)/2]}]; Table[a[n], {n, 0, 30}]

Prog

(PARI) sum(k=0, n\2, k*binomial(n+k, k)*binomial(n, n-2*k)) + sum(k=0, (n+1)\2, k*binomial(n+k-1, k-1)*binomial(n, n-2*k+1))

Crossrefs

The denominator is A000984.

Sequence in context: A102285 A218985 A129162 * A290924 A026760 A064914

Adjacent sequences: A167657 A167658 A167659 * A167661 A167662 A167663

Keyword

nonn,frac

Author

Roger L. Bagula, Nov 08 2009

Extensions

Edited by Charles R Greathouse IV, Nov 09 2009

Status

approved