A131065 -id:A131065 - cp's OEIS frontend

A153491 Triangle T(n,m)= 11*binomial(n,k) - 8 read by rows, 0<=k<=n.

3, 3, 3, 3, 14, 3, 3, 25, 25, 3, 3, 36, 58, 36, 3, 3, 47, 102, 102, 47, 3, 3, 58, 157, 212, 157, 58, 3, 3, 69, 223, 377, 377, 223, 69, 3, 3, 80, 300, 608, 762, 608, 300, 80, 3, 3, 91, 388, 916, 1378, 1378, 916, 388, 91, 3, 3, 102, 487, 1312

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 27 2008

Row sums are 3(n+1) + 11*(2^n - n - 1) = 3, 6, 20, 56, 136,...

			3;
3,3;
3,14,3;
3,25,25,3;
3,36,58,36,3;
3,47,102,102,47,3;
3,58,157,212,157,58,3;
3,69,223,377,377,223,69,3;
3,80,300,608,762,608,300,80,3;

G. C. Greubel, Table of n, a(n) for n = 0..1034

Cf. A131065

T[n_, m_] = If[n == 1, 3, If[m == 0 || m == n, 3, 11*Binomial[n, k] - 8]]
a1 = Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}];
Flatten[a1]

Definition and terms regularized. - R. J. Mathar, Jul 11 2012

A176203 Triangle read by rows: T(n, k) = 16*binomial(n, k) - 15.

Original entry on oeis.org

1, 1, 1, 1, 17, 1, 1, 33, 33, 1, 1, 49, 81, 49, 1, 1, 65, 145, 145, 65, 1, 1, 81, 225, 305, 225, 81, 1, 1, 97, 321, 545, 545, 321, 97, 1, 1, 113, 433, 881, 1105, 881, 433, 113, 1, 1, 129, 561, 1329, 2001, 2001, 1329, 561, 129, 1, 1, 145, 705, 1905, 3345, 4017, 3345, 1905, 705, 145, 1

Offset: 0

Roger L. Bagula, Apr 11 2010

This sequence belongs to the class defined by T(n, m, q) = 2*T(n, m, q-1) - 1. The first few q values gives the sequences: A007318 (q=0), A109128 (q=1), A131061 (q=2), A168625 (q=3), this sequence (q=4).
Row sums are: {1, 2, 19, 68, 181, 422, 919, 1928, 3961, 8042, 16219, ...}.
Former title: A recursive symmetrical triangular sequence:q=4: t(n, m, q) = 2*t(n, m, q-1) - 1. - G. C. Greubel, Mar 12 2020

			Triangle begins as:
  1;
  1,   1;
  1,  17,   1;
  1,  33,  33,    1;
  1,  49,  81,   49,    1;
  1,  65, 145,  145,   65,    1;
  1,  81, 225,  305,  225,   81,    1;
  1,  97, 321,  545,  545,  321,   97,    1;
  1, 113, 433,  881, 1105,  881,  433,  113,   1;
  1, 129, 561, 1329, 2001, 2001, 1329,  561, 129,   1;
  1, 145, 705, 1905, 3345, 4017, 3345, 1905, 705, 145, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Sequence m*binomial(n,k) - (m-1): A007318 (m=1), this sequence (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), A168625 (m=8), this sequence (m=16).

[16*Binomial(n, k) -15: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020

A176203:= (n,k) -> 16*binomial(n, k) -15; seq(seq(A176203(n, k), k = 0..n), n = 0.. 12); # G. C. Greubel, Mar 12 2020

T[n_, m_, q]:= 2^q*(Binomial[n, m] -1) + 1; Table[T[n,m,4], {n,0,12}, {m,0,n} ]//Flatten (* modified by G. C. Greubel, Mar 12 2020 *)
Table[16*Binomial[n, k] -15, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)

[[16*binomial(n, k) -15 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

T(n, m, q) = 2*T(n, m, q-1) - 1, with T(n, m, 0) = binomial(n, m) and q = 4.
From G. C. Greubel, Mar 12 2020: (Start)
T(n, k, q) = 2^q * binomial(n, k) - (2^q - 1), with q = 4.
Sum_{k=0..n} T(n, k, q) = 2^(n + q) - (n + 1)*(2^q - 1) (row sums). (End)

Edited by G. C. Greubel, Mar 12 2020

cp's OEIS Frontend

A153491 Triangle T(n,m)= 11*binomial(n,k) - 8 read by rows, 0<=k<=n.

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