A137398 -id:A137398 - cp's OEIS frontend

A188475 a(n) = (2n^3 + 3n^2 + n + 3)/3.

1, 3, 11, 29, 61, 111, 183, 281, 409, 571, 771, 1013, 1301, 1639, 2031, 2481, 2993, 3571, 4219, 4941, 5741, 6623, 7591, 8649, 9801, 11051, 12403, 13861, 15429, 17111, 18911, 20833, 22881, 25059, 27371, 29821, 32413, 35151, 38039, 41081, 44281, 47643

Offset: 0

Paul Barry, Apr 01 2011

Hankel transform of A137398(n+1) (conjecture).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[(2*n^3+3*n^2+n+3)/3: n in [0..50]]; // Vincenzo Librandi, Nov 25 2011

A188475:=n->(2*n^3+3*n^2+n+3)/3; seq(A188475(n),n=0..100); # Wesley Ivan Hurt, Nov 11 2013

LinearRecurrence[{4,-6,4,-1},{1,3,11,29},100] (* Vincenzo Librandi, Nov 25 2011 *)

G.f.: (1 - x + 5*x^2 - x^3)/(1-x)^4.

a(n) = A006331(n) + 1. - Bruno Berselli, Nov 14 2011

A188474 A generalized Deutsch triangle.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 10, 10, 1, 1, 16, 40, 16, 1, 1, 23, 102, 102, 23, 1, 1, 31, 209, 393, 209, 31, 1, 1, 40, 376, 1122, 1122, 376, 40, 1, 1, 50, 620, 2656, 4296, 2656, 620, 50, 1, 1, 61, 960, 5536, 13100, 13100, 5536, 960, 61, 1, 1, 73, 1417, 10522, 34036, 50180, 34036, 10522, 1417, 73, 1

Offset: 0

Paul Barry, Apr 01 2011

Member r=2 of the family of "Pascal-like" triangles with T(n,k)=sum{j=0..n-k+1, (j/(n+2-j))*C(n+2-j,n-k+1)*C(n+2-j,k+1)*r^(j-1)}.
The Deutsch triangle A100754 corresponds to r=1.
Row sums are A137398(n+1) (conjecture). Diagonal sums are A188476.

			Triangle begins
1,
1, 1,
1, 5, 1,
1, 10, 10, 1,
1, 16, 40, 16, 1,
1, 23, 102, 102, 23, 1,
1, 31, 209, 393, 209, 31, 1,
1, 40, 376, 1122, 1122, 376, 40, 1,
1, 50, 620, 2656, 4296, 2656, 620, 50, 1,
1, 61, 960, 5536, 13100, 13100, 5536, 960, 61, 1,
1, 73, 1417, 10522, 34036, 50180, 34036, 10522, 1417, 73, 1

T(n,k)=sum{j=0..n-k+1, (j/(n+2-j))*C(n+2-j,n-k+1)*C(n+2-j,k+1)*2^(j-1)}.

cp's OEIS Frontend

A188475 a(n) = (2n^3 + 3n^2 + n + 3)/3.

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A188474 A generalized Deutsch triangle.

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cp's OEIS Frontend

A188475 a(n) = (2*n^3 + 3*n^2 + n + 3)/3.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

Formula

A188474 A generalized Deutsch triangle.

Views

Author

Keywords

Comments

Examples

Formula

A188475 a(n) = (2n^3 + 3n^2 + n + 3)/3.