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 A055005 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n. 0
 1, 0, 0, 6, 63, 306, 1038, 2844, 6750, 14437, 28521, 52911, 93258, 157509, 256581, 405171, 622719, 934542, 1373158, 1979820, 2806281, 3916812, 5390496, 7323822, 9833604, 13060251, 17171415, 22366045, 28878876, 36985383, 47007231 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..30. Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1). FORMULA Number of nonnegative integer p X q matrices with no zero rows or columns and with sum of elements equal to n is Sum_{k=0...q} (-1)^k*C(q, k)*m(p, q-k, n) where m(p, q, n)=Sum_{k=0..p} (-1)^k*C(p, k)*C((p-k)*q+n-1, n). For p=q=3 we get a(n)=C(n + 8, 8) - 6*C(n + 5, 5) + 9*C(n + 3, 3) + 6*C(n + 2, 2) - 18*C(n + 1, 1) + 9=(1/8!)*(n^8 + 36*n^7 + 546*n^6 + 2520*n^5 - 7791*n^4 - 43596*n^3 + 148364*n^2 - 140400*n + 40320). G.f.: -(9*x^8-54*x^7+132*x^6-171*x^5+135*x^4-78*x^3+36*x^2-9*x+1) / (x-1)^9. - Colin Barker, Jul 13 2013 PROG (PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1; 1, -9, 36, -84, 126, -126, 84, -36, 9]^n*[1; 0; 0; 6; 63; 306; 1038; 2844; 6750])[1, 1] \\ Charles R Greathouse IV, Aug 14 2023 CROSSREFS Cf. A054688, A054974, A054975. Sequence in context: A222079 A158987 A339239 * A027811 A027950 A184447 Adjacent sequences: A055002 A055003 A055004 * A055006 A055007 A055008 KEYWORD easy,nonn AUTHOR Vladeta Jovovic, May 30 2000 STATUS approved

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Last modified December 8 06:57 EST 2023. Contains 367662 sequences. (Running on oeis4.)