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A116392 Riordan array (1/sqrt(1-2x-3x^2), 1/sqrt(1-2x-3x^2)-1). 5
1, 1, 1, 3, 4, 1, 7, 13, 7, 1, 19, 42, 32, 10, 1, 51, 131, 128, 60, 13, 1, 141, 406, 475, 292, 97, 16, 1, 393, 1247, 1685, 1267, 561, 143, 19, 1, 1107, 3814, 5800, 5112, 2804, 962, 198, 22, 1, 3139, 11623, 19540, 19624, 12748, 5464, 1522, 262, 25, 1, 8953, 35334 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

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

FORMULA

Number triangle T(n,k) = Sum_{j=0..n} C(n,j)*A116389(j,k).

EXAMPLE

Triangle begins:

   1;

   1,   1;

   3,   4,   1;

   7,  13,   7,  1;

  19,  42,  32, 10,  1;

  51, 131, 128, 60, 13, 1;

MATHEMATICA

t[n_, k_]:= Sum[(-1)^(k-j)*Binomial[k, j]*Sum[4^r*Binomial[r+(j-1)/2, r]* Binomial[j, n-2*r], {r, 0, Floor[n/2]}], {j, 0, k}]; Table[Sum[Binomial[n, j]*t[j, k], {j, 0, n}] {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 23 2019 *)

PROG

(PARI) t(n, k) = sum(j=0, k, sum(r=0, floor(n/2), (-1)^(k-j)*4^r* binomial(k, j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) ));

T(n, k) = sum(j=0, n, binomial(n, j)*t(j, k)); \\ G. C. Greubel, May 23 2019

(MAGMA) [[(&+[ Binomial(n, m)*(&+[ (&+[ Round((-1)^(k-j)*4^r* Binomial(k, j)*Binomial(j, m-2*r)*Gamma(r+(j+1)/2)/(Factorial(r)*Gamma((j+1)/2))) : r in [0..Floor(n/2)]]) : j in [0..k]]): m in [0..n]]) : k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 23 2019

(Sage) [[sum(binomial(n, m)*sum( sum( (-1)^(k-j)*4^r* binomial(k, j)* binomial(r+(j-1)/2, r)*binomial(j, m-2*r) for r in (0..floor(n/2))) for j in (0..k)) for m in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 23 2019

CROSSREFS

Row sums are A115967. Diagonal sums are A116394.

Sequence in context: A075052 A111516 A210636 * A324559 A174607 A326503

Adjacent sequences:  A116389 A116390 A116391 * A116393 A116394 A116395

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Feb 12 2006

STATUS

approved

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Last modified December 7 22:29 EST 2019. Contains 329850 sequences. (Running on oeis4.)