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A365962 Triangle read by rows: coefficients in expansion of Asveld's polynomials Pi_j(x).

%I A365962 #22 Apr 27 2025 00:46:12
%S A365962 1,3,1,10,6,1,48,30,9,1,312,192,60,12,1,2520,1560,480,100,15,1,24480,
%T A365962 15120,4680,960,150,18,1,277200,171360,52920,10920,1680,210,21,1,
%U A365962 3588480,2217600,685440,141120,21840,2688,280,24,1,52254720,32296320,9979200,2056320,317520,39312,4032,360,27,1
%N A365962 Triangle read by rows: coefficients in expansion of Asveld's polynomials Pi_j(x).
%H A365962 Winston de Greef, <a href="/A365962/b365962.txt">Table of n, a(n) for n = 0..11324</a>
%H A365962 P. R. J. Asveld, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-4/asveld.pdf">Another family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 361-364.
%F A365962 E.g.f. of column k: (1+x)^2*x^k / ((1-x-x^2)*k!), k >= 0.
%F A365962 T(n,n) = 1 and T(n,k) = n!/k!*Fibonacci(n-k+3), n > k >= 0.
%F A365962 T(n,k) = n!/k!*Sum_{j=k..n} Fibonacci(j-k+1)*binomial(2,n-j).
%F A365962 T(n,k) = n!/k!*Sum_{j=k..n} (Fibonacci(j-k)+(-1)^(j-k))*binomial(3,n-j).
%F A365962 Recurrence: T(n,0) = A005921(n) and T(n,k) = n*T(n-1,k-1) / k, n >= k >= 1.
%F A365962 T(n,k) = Sum_{j=k..n} Stirling2(j,k)*(Sum_{i=j..n} Stirling1(n,i)*A341725(i,j)).
%F A365962 Sum_{j=k..n} (-1)^(n-j)*(n-j+1)!*binomial(n,j)*T(j,k) = A039948(n,k).
%e A365962 Triangle begins:
%e A365962       1,
%e A365962       3,     1,
%e A365962      10,     6,    1,
%e A365962      48,    30,    9,   1,
%e A365962     312,   192,   60,  12,   1,
%e A365962    2520,  1560,  480, 100,  15,  1,
%e A365962   24480, 15120, 4680, 960, 150, 18, 1,
%e A365962   ...
%p A365962 T := proc(n, k) option remember;if k = n then 1  else (n!/k!*combinat[fibonacci](n-k+3)) fi end: seq(print(seq(T(n, k), k = 0..n)), n=0..9);
%p A365962 # second Maple program:
%p A365962 T := (n,k) -> add(Stirling2(j, k)*add(Stirling1(n, i)*A341725(i, j), i = j .. n), j = k .. n): seq(print(seq(T(n, k), k = 0 .. n)), n = 0 .. 9);
%o A365962 (PARI) T(n,k) = n!/k!*sum(j=k, n, fibonacci(j-k+1)*binomial(2,n-j)) \\ _Winston de Greef_, Oct 21 2023
%o A365962 (PARI) T(n,k) = if(n == k, 1, n!/k!*fibonacci(n-k+3)) \\ _Winston de Greef_, Oct 21 2023
%Y A365962 Cf. A000045, A005921 (col 0), A005922 (col 1), A039948, A341725.
%K A365962 nonn,tabl
%O A365962 0,2
%A A365962 _Mélika Tebni_, Sep 23 2023