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A131239 Triangle, T(n,k) = 3*A007318(n,k) - 2*A046854(n,k), read by rows. 2
1, 1, 1, 1, 4, 1, 1, 5, 7, 1, 1, 8, 12, 10, 1, 1, 9, 24, 22, 13, 1, 1, 12, 33, 52, 35, 16, 1, 1, 13, 51, 85, 95, 51, 19, 1, 1, 16, 64, 148, 180, 156, 70, 22, 1, 1, 17, 88, 212, 348, 336, 238, 92, 25, 1, 1, 20, 105, 320, 560, 714, 574, 344, 117, 28, 1, 1, 21, 135, 425, 920, 1274, 1330, 918, 477, 145, 31, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums = A074878: (1, 2, 6, 14, 32, 70, 239,...).

A131238 = 2*A007318 - A046854.

LINKS

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

FORMULA

T(n,k) = 3*A007318(n,k) - 2*A046854(n,k) as infinite lower triangular matrices.

T(n,k) = 3*binomial(n,k) - 2*binomial(floor((n+k)/2), k). - G. C. Greubel, Jul 12 2019

EXAMPLE

First few rows of the triangle are:

  1;

  1,  1;

  1,  4,  1;

  1,  5,  7,  1;

  1,  8, 12, 10,  1;

  1,  9, 24, 22, 13,  1;

  1, 12, 33, 52, 35, 16, 1; ...

MATHEMATICA

With[{B=Binomial}, Table[3*B[n, k] - 2*B[Floor[(n+k)/2], k], {n, 0, 12}, {k, 0, n}]]//Flatten (* G. C. Greubel, Jul 12 2019 *)

PROG

(PARI) b=binomial; T(n, k) = 3*b(n, k) - 2*b((n+k)\2, k);

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 12 2019

(MAGMA) B:=Binomial; [3*B(n, k) - 2*B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2019

(Sage) b=binomial; [[3*b(n, k) - 2*b(floor((n+k)/2), k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 12 2019

(GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> 3*B(n, k) - 2*B(Int((n+k)/2), k) ))); # G. C. Greubel, Jul 12 2019

CROSSREFS

Cf. A007318, A046854, A131238, A074878.

Sequence in context: A171142 A174037 A173077 * A114033 A334426 A209417

Adjacent sequences:  A131236 A131237 A131238 * A131240 A131241 A131242

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jun 21 2007

EXTENSIONS

More terms added by G. C. Greubel, Jul 12 2019

STATUS

approved

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Last modified June 21 19:59 EDT 2021. Contains 345365 sequences. (Running on oeis4.)