A137398 -id:A137398

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A188475

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

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

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

MAPLE

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

MATHEMATICA

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

PROG

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

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Apr 01 2011

STATUS

approved

A188474

A generalized Deutsch triangle.

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

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)}.

EXAMPLE

Triangle begins

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