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A144403 Triangle T(n, k) = binomial(n, k)^2 - binomial(n, k) - 1, read by rows. 1
-1, -1, -1, -1, 1, -1, -1, 5, 5, -1, -1, 11, 29, 11, -1, -1, 19, 89, 89, 19, -1, -1, 29, 209, 379, 209, 29, -1, -1, 41, 419, 1189, 1189, 419, 41, -1, -1, 55, 755, 3079, 4829, 3079, 755, 55, -1, -1, 71, 1259, 6971, 15749, 15749, 6971, 1259, 71, -1, -1, 89, 1979, 14279, 43889, 63251, 43889, 14279, 1979, 89, -1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = binomial(n, k)^2 - binomial(n, k) - 1.

Sum_{k=0..n} T(n,k) = Binomial(2*n, n) - 2^n - n - 1. - G. C. Greubel, Mar 27 2021

EXAMPLE

Triangle begins as:

  -1;

  -1, -1;

  -1,  1,   -1;

  -1,  5,    5,    -1;

  -1, 11,   29,    11,    -1;

  -1, 19,   89,    89,    19,    -1;

  -1, 29,  209,   379,   209,    29,    -1;

  -1, 41,  419,  1189,  1189,   419,    41,    -1;

  -1, 55,  755,  3079,  4829,  3079,   755,    55,   -1;

  -1, 71, 1259,  6971, 15749, 15749,  6971,  1259,   71, -1;

  -1, 89, 1979, 14279, 43889, 63251, 43889, 14279, 1979, 89, -1;

MAPLE

A144403:= (n, k)-> binomial(n, k)^2 - binomial(n, k) - 1;

seq(seq(A144403(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 27 2021

MATHEMATICA

Table[Binomial[n, m]^2 -Binomial[n, m] -1, {n, 0, 12}, {m, 0, n}]//Flatten

PROG

(Magma) [Binomial(n, k)^2 - Binomial(n, k) - 1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2021

(Sage) flatten([[binomial(n, k)^2 - binomial(n, k) - 1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 27 2021

CROSSREFS

Cf. A000984.

Sequence in context: A046568 A046571 A172349 * A188587 A174119 A156696

Adjacent sequences:  A144400 A144401 A144402 * A144404 A144405 A144406

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

EXTENSIONS

Edited by G. C. Greubel, Mar 27 2021

STATUS

approved

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Last modified July 23 11:21 EDT 2021. Contains 346259 sequences. (Running on oeis4.)