A051743 -id:A051743 - cp's OEIS frontend

A105450 a(n) = binomial(n+5,6) + binomial(n+3,3) + binomial(n+2,3) + binomial(n-1,1).

0, 6, 22, 60, 142, 305, 607, 1134, 2008, 3396, 5520, 8668, 13206, 19591, 28385, 40270, 56064, 76738, 103434, 137484, 180430, 234045, 300355, 381662, 480568, 600000, 743236, 913932, 1116150, 1354387, 1633605, 1959262, 2337344, 2774398, 3277566, 3854620

Offset: 0

D. G. Rogers, May 07 2005

Number of directed column-convex polyominoes with perimeter 2(n+4) having n cells in the foundational column.

A051743 and this sequence form successive diagonals in an array that has as row sums the sequence A006027.

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A051743, A006027.

Table[Binomial[n+5,6]+Binomial[n+3,3]+Binomial[n+2,3]+ Binomial[n-1,1],{n,0,50}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,6,22,60,142,305,607},51] (* Harvey P. Dale, Jun 28 2011 *)

a(n)=n*(n^5+15*n^4+85*n^3+465*n^2+1354*n+2400)/720 \\ Charles R Greathouse IV, Oct 16 2015

a(0)=0, a(1)=6, a(2)=22, a(3)=60, a(4)=142, a(5)=305, a(6)= 607, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)- 7*a(n-6)+a(n-7). - Harvey P. Dale, Jun 28 2011

G.f.: (2*x^6-11*x^5+26*x^4-32*x^3+20*x^2-6*x)/(x-1)^7. - Harvey P. Dale, Jun 28 2011

A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 2, 0, 1, 4, 6, 5, 2, 0, 1, 5, 10, 11, 7, 2, 1, 1, 6, 15, 21, 18, 9, 3, 0, 1, 7, 21, 36, 39, 27, 12, 3, 0, 1, 8, 28, 57, 75, 66, 39, 15, 3, 1, 1, 9, 36, 85, 132, 141, 105, 54, 18, 4, 0, 1, 10, 45, 121, 217, 273, 246, 159, 72, 22, 4, 0, 1, 11, 55, 166

Offset: 0

Alford Arnold, May 05 2006

The fourth diagonal is 1, 2, 5, 11, 21, ..., which is 1 + A000292. The fifth diagonal is 0, 2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, ..., which is A051743.
The array A007318 is generated by placing A000012 on both edges with the same Pascal-like recurrence, and the array A059259 uses edges defined by A000012 and A059841. - R. J. Mathar, Jan 21 2008
From Michael A. Allen, Nov 30 2021: (Start)
T(n,n-k) is the (n,k)-th entry of the (1/(1-x^3), x/(1-x)) Riordan array.
Sums of rows give A077947.
Sums of antidiagonals give A079962. (End)

			The table begins
  1
  1  0
  1  1  0
  1  2  1  1
  1  3  3  2  0
  1  4  6  5  2  0
  1  5 10 11  7  2  1
  1  6 15 21 18  9  3  0

Cf. A000292, A079978, A008620, A079998, A051743, A077947.

Generalization of A007318, A059259.

A000012 := proc(n) 1 ; end: A079978 := proc(n) if n mod 3 = 0 then 1; else 0 ; fi ; end: A118923 := proc(n,k) if k = 0 then A000012(n); elif k = n then A079978(n) ; else A118923(n-1,k)+A118923(n-1,k-1) ; fi ; end: for n from 0 to 15 do for k from 0 to n do printf("%d, ",A118923(n,k)) ; od: od: # R. J. Mathar, Jan 21 2008

Flatten@Table[CoefficientList[Series[1/((1 + x*y + x^2*y^2)(1 - x - x*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n + 1, k + 1]], {n, 0, 11}, {k, 0, n}] (* Michael A. Allen, Nov 30 2021 *)

From Michael A. Allen, Nov 30 2021: (Start)
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/3)} binomial(n-3*j,n-k)/(n-3*j).
G.f.: 1/((1+x*y+(x*y)^2)*(1-x-x*y)). (End)

Edited and extended by R. J. Mathar, Jan 21 2008

Offset changed by Michael A. Allen, Nov 30 2021

cp's OEIS Frontend

A105450 a(n) = binomial(n+5,6) + binomial(n+3,3) + binomial(n+2,3) + binomial(n-1,1).

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