A132729 -id:A132729 - cp's OEIS frontend

A132730 Row sums of triangle A132729.

1, 2, 3, 8, 21, 50, 111, 236, 489, 998, 2019, 4064, 8157, 16346, 32727, 65492, 131025, 262094, 524235, 1048520, 2097093, 4194242, 8388543, 16777148, 33554361, 67108790, 134217651, 268435376, 536870829

Offset: 0

Gary W. Adamson, Aug 26 2007

			a(4) = 21 = sum of row 4 terms of triangle A132729: (1 + 5 + 9 + 5 + 1).
a(3) = 8 = (1, 3, 3, 1) dot (1, 1, 0, 4) = (1 + 3 + 0 + 4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A132729.

[1] cat [2^(n+1) -3*n +1: n in [0..30]]; // G. C. Greubel, Feb 14 2021

LinearRecurrence[{4,-5,2},{1,2,3,8},30] (* Harvey P. Dale, Dec 30 2015 *)
Table[2^(n+1) -3*n +1 -2*Boole[n==0], {n,0,30}] (* G. C. Greubel, Feb 14 2021 *)

[1]+[2^(n+1) -3*n +1 for n in (1..30)] # G. C. Greubel, Feb 14 2021

Binomial transform of [1, 1, 0, 4, 0, 4, 0, 4, ...].
a(n) = 2^(n+1) - 3*n + 1, for n > 0. - R. J. Mathar, Apr 04 2012
From G. C. Greubel, Feb 14 2021: (Start)
G.f.: (1 - 2*x + 4*x^3)/((1-x)^2 * (1-2*x)).
E.g.f.: -2 + (1-3*x)*exp(x) + 2*exp(2*x). (End)

A132737 Triangle T(n,k) = 2*binomial(n,k) + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 7, 7, 1, 1, 9, 13, 9, 1, 1, 11, 21, 21, 11, 1, 1, 13, 31, 41, 31, 13, 1, 1, 15, 43, 71, 71, 43, 15, 1, 1, 17, 57, 113, 141, 113, 57, 17, 1, 1, 19, 73, 169, 253, 253, 169, 73, 19, 1, 1, 21, 91, 241, 421, 505, 421, 241, 91, 21, 1, 1, 23, 111, 331, 661, 925, 925, 661, 331, 111, 23, 1

Offset: 0

Gary W. Adamson, Aug 26 2007

			First few rows of the triangle are:
  1;
  1,  1;
  1,  5,  1;
  1,  7,  7,  1;
  1,  9, 13,  9,  1;
  1, 11, 21, 21, 11,  1;
  1, 13, 31, 41, 31, 13,  1;
  1, 15, 43, 71, 71, 43, 15, 1;
  ...

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

Cf. A132735, A132738.

Sequences of the form 2*binomial(n,k) + q: A132729 (q=-3), A132731 (q=-2), A109128 (q=-1), A132046 (q=0), this sequence (q=1).

A132737:= func< n,k | k eq 0 or k eq n select 1 else 2*Binomial(n,k) +1 >;
[A132737(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 15 2021

T[n_, k_]:= If[k==0 || k==n, 1, 2*Binomial[n,k] +1];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)

def A132737(n,k): return 1 if (k==0 or k==n) else 2*binomial(n,k) + 1
flatten([[A132737(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 15 2021

T(n, k) = 2*A132735(n, k) - 1, an infinite lower triangular matrix.
T(n,0) = T(n,n) = 1; otherwise T(n,k) = 2*C(n,k) + 1. - Franklin T. Adams-Watters, Jul 06 2009
Sum_{k=0..n} T(n, k) = 2^(n+1) + n - 3 + 2*[n=0] = A132738(n). - G. C. Greubel, Feb 15 2021

Extended by Franklin T. Adams-Watters, Jul 06 2009

cp's OEIS Frontend

A132730 Row sums of triangle A132729.

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