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 A115990 Riordan array (1/sqrt(1-2*x-3*x^2), ((1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2). 4
 1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 19, 13, 8, 4, 1, 51, 35, 22, 12, 5, 1, 141, 96, 61, 35, 17, 6, 1, 393, 267, 171, 101, 53, 23, 7, 1, 1107, 750, 483, 291, 160, 77, 30, 8, 1, 3139, 2123, 1373, 839, 476, 244, 108, 38, 9, 1, 8953, 6046, 3923, 2423, 1406, 752, 360, 147, 47, 10 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS First column is central trinomial coefficients A002426. Second column is number of directed animals of size n+1, A005773(n+1). Row sums are A005717 (number of horizontal steps in all Motzkin paths of length n). First column has e.g.f. exp(x) I_0(2x). Row sums have e.g.f. dif(exp(x) I_1(2x),x). Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/2). LINKS G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened FORMULA Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j). EXAMPLE Triangle begins 1; 1, 1; 3, 2, 1; 7, 5, 3, 1; 19, 13, 8, 4, 1; 51, 35, 22, 12, 5, 1; 141, 96, 61, 35, 17, 6, 1; MAPLE A115990 := proc(n, k) add(binomial(n-k, j-k)*binomial(j, n-j), j=0..n) ; end proc: seq(seq(A115990(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Jun 25 2023 MATHEMATICA Table[Sum[ Binomial[n-k, j-k]*Binomial[j, n-j], {j, 0, n}], {n, 0, 10}, {k, 0, n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *) PROG (PARI) {T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j))}; \\ G. C. Greubel, May 09 2019 (Magma) [[(&+[Binomial(n-k, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019 (Sage) [[sum(binomial(n-k, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019 (GAP) Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)) ))); # G. C. Greubel, May 09 2019 CROSSREFS Cf. A115991, A005773 (k=1), A025566 (k=2), A035045 (k=3), A152948 (diag. n=k+2), . Sequence in context: A364855 A105531 A129689 * A350571 A277919 A094531 Adjacent sequences: A115987 A115988 A115989 * A115991 A115992 A115993 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Feb 10 2006 STATUS approved

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Last modified December 11 14:52 EST 2023. Contains 367727 sequences. (Running on oeis4.)