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 A115991 Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j)*2^(n-j). 2
 1, 1, 1, 5, 3, 1, 13, 9, 5, 1, 49, 31, 17, 7, 1, 161, 105, 61, 29, 9, 1, 581, 371, 217, 111, 45, 11, 1, 2045, 1313, 781, 417, 189, 65, 13, 1, 7393, 4719, 2825, 1551, 753, 303, 89, 15, 1, 26689, 17041, 10277, 5757, 2921, 1289, 461, 117, 17, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS First column is A084601 with e.g.f. exp(x) Bessel_I(0,2*sqrt(2)x). Row sums are A098518(n+1) with e.g.f. dif(exp(x) Bessel_I(1,2*sqrt(2)x)/sqrt(2)). Riordan array (1/sqrt(1-2*x-7*x^2), (1+x-sqrt(1-2*x-7*x^2))/2). LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened EXAMPLE Triangle begins as: 1; 1, 1; 5, 3, 1; 13, 9, 5, 1; 49, 31, 17, 7, 1; 161, 105, 61, 29, 9, 1; 581, 371, 217, 111, 45, 11, 1; MAPLE A115991 := proc(n, k) add(binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j), j=0..n) ; end proc: seq(seq(A115991(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]*2^(n-j), {j, 0, n}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 09 2019 *) PROG (PARI) {T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j)*2^(n-j))}; \\ G. C. Greubel, May 09 2019 (Magma) [[(&+[Binomial(n-k, j-k)*Binomial(j, n-j)*2^(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)*2^(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)*2^(n-j)) ))); # G. C. Greubel, May 09 2019 CROSSREFS Cf. A084601 (k=0), A098518. Sequence in context: A229958 A157891 A173644 * A352136 A143410 A114344 Adjacent sequences: A115988 A115989 A115990 * A115992 A115993 A115994 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Feb 10 2006 STATUS approved

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Last modified December 9 17:15 EST 2023. Contains 367693 sequences. (Running on oeis4.)