A116392 Riordan array (1/sqrt(1-2*x-3*x^2), 1/sqrt(1-2*x-3*x^2) -1).

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

Offset

0,4

Comments

Triangle, read by rows, given by [1, 2, -1, -1, 2, 1/2, 1/2, 2, -1, -1, 2, 1/2, 1/2, 2, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 11 2020

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;

Maple

# The function RiordanSquare is defined in A321620.
RiordanSquare(1/sqrt(1 - 2*x - 3*x^2), 10); # Peter Luschny, Feb 15 2020

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.

Cf. A321620.

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