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A039948 A triangle related to A000045 (Fibonacci numbers).

%I A039948 #27 Jan 05 2025 19:51:35
%S A039948 1,1,1,4,2,1,18,12,3,1,120,72,24,4,1,960,600,180,40,5,1,9360,5760,
%T A039948 1800,360,60,6,1,105840,65520,20160,4200,630,84,7,1,1370880,846720,
%U A039948 262080,53760,8400,1008,112,8,1,19958400,12337920,3810240,786240,120960,15120,1512,144,9,1
%N A039948 A triangle related to A000045 (Fibonacci numbers).
%H A039948 Seiichi Manyama, <a href="/A039948/b039948.txt">Rows n = 0..139, flattened</a>
%H A039948 P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-4/asveld.pdf">Fibonacci-like differential equations with a polynomial nonhomogeneous term</a>, Fib. Quart. 27 (1989), 303-309.
%F A039948 T(n, m) = n!*Fibonacci(n-m+1)/m!, n >= m >= 0.
%F A039948 T(n, 0) = A005442(n).
%F A039948 T(n, 1) = A005443(n).
%F A039948 E.g.f. for column m: x^m/(m!*(1-x-x^2)), m >= 0.
%F A039948 From _G. C. Greubel_, Nov 20 2022: (Start)
%F A039948 T(n, n-1) = A000027(n).
%F A039948 T(n, n-2) = 4*A000217(n-1), n >= 2.
%F A039948 T(n, n-3) = 18*A000292(n-2), n >= 3.
%F A039948 T(n, n-4) = 5! * A000332(n), n >= 4.
%F A039948 T(n, n-5) = 8 * 5! * A000389(n), n >= 5.
%F A039948 T(n, n-6) = 13 * 6! * A000579(n), n >= 6.
%F A039948 T(n, n-7) = 21 * 7! * A000580(n), n >= 7.
%F A039948 T(n, n-8) = 34 * 8! * A000581(n), n >= 8.
%F A039948 T(n, n-9) = 55 * 9! * A000582(n), n >= 9.
%F A039948 T(n, n-10) = 89 * 10! * A001287(n), n >= 10.
%F A039948 T(n, n-11) = 12 * 12! * A001288(n), n >= 11.
%F A039948 T(n, n-12) = 233 * 12! * A010965(n), n >= 12.
%F A039948 T(n, n-13) = 89 * 13! * A010966(n), n >= 13.
%F A039948 Sum_{k=0..n} T(n, k) = A110313(n). (End)
%e A039948 Triangle begins :
%e A039948     1;
%e A039948     1,   1;
%e A039948     4,   2,   1;
%e A039948    18,  12,   3,  1;
%e A039948   120,  72,  24,  4, 1;
%e A039948   960, 600, 180, 40, 5, 1;
%e A039948 ... - _Philippe Deléham_, Nov 08 2011
%t A039948 T[n_,k_]:= (n!/k!)*Fibonacci[n-k+1];
%t A039948 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2022 *)
%o A039948 (Magma) [(Factorial(n)/Factorial(k))*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2022
%o A039948 (SageMath)
%o A039948 def A039948(n, k): return factorial(n-k)*binomial(n,k)*fibonacci(n-k+1)
%o A039948 flatten([[A039948(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Nov 20 2022
%Y A039948 Cf. A000045, A000142, A005442, A005443, A110313 (row sums).
%Y A039948 Diagonals include: A000027, A000217, A000292, A000332, A000389, A000579, A000580, A000581, A000582, A001287, A001288, A010965, A010966.
%K A039948 nonn,tabl
%O A039948 0,4
%A A039948 _Wolfdieter Lang_