A073404 -id:A073404 - cp's OEIS frontend

A073403 Coefficient triangle of polynomials (falling powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073404.

1, 12, 36, 120, 888, 1536, 1152, 15168, 62592, 80448, 10944, 222336, 1600704, 4813056, 5068800, 103680, 2992896, 32811264, 169917696, 413351424, 375598080, 981504, 38112768, 587976192, 4592982528

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..

The k-th convolution of U0(n) := A002605(n), n>= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*12^k), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073404(k,m).

			k=2: U2(n)=(2*(36+12*n)*(n+1)*U0(n+1)+2*(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
1; 12,36; 120,888,1536; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows.

Cf. A002605, A073387, A073404.

Recursion for row polynomials defined in the comments: see A073405.

A073387 Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 16, 16, 6, 1, 44, 56, 30, 8, 1, 120, 188, 128, 48, 10, 1, 328, 608, 504, 240, 70, 12, 1, 896, 1920, 1872, 1080, 400, 96, 14, 1, 2448, 5952, 6672, 4512, 2020, 616, 126, 16, 1, 6688, 18192, 23040, 17856, 9352, 3444, 896, 160, 18, 1

Offset: 0

Wolfdieter Lang, Aug 02 2002

The g.f. for the row polynomials P(n,x) = Sum_{m=0..n} T(n,m)*x^m is 1/(1-(2+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.

T(n, k) is the number of words of length n over {0,1,2,3} having k letters 3 and avoiding runs of odd length for the letters 0,1. - Milan Janjic, Jan 14 2017

			Lower triangular matrix, T(n,k), n >= k >= 0, else 0:
    1;
    2,    1;
    6,    4,    1;
   16,   16,    6,    1;
   44,   56,   30,    8,   1;
  120,  188,  128,   48,  10,   1;
  328,  608,  504,  240,  70,  12,   1;
  896, 1920, 1872, 1080, 400,  96,  14,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Wolfdieter Lang, First 10 rows.

Cf. A002605, A007482 (row sums), A053121, A073403, A073404.
Cf. A001045, A005563, A015518, A159920.
Columns: A002605 (k=0), A073388 (k=1), A073389 (k=2), A073390 (k=3), A073391 (k=4), A073392 (k=5), A073393 (k=6), A073394 (k=7), A073397 (k=8), A073398 (k=9).

A073387:= func< n,k | (&+[2^(n-k-j)*Binomial(n-j,k)*Binomial(n-k-j,j): j in [0..Floor((n-k)/2)]]) >;
[A073387(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 03 2022

T := (n,k) -> `if`(n=0,1,2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # Peter Luschny, Apr 25 2016

T[n_, k_]:=T[n,k]=Sum[2^(n-k-j)*Binomial[n-j,k]*Binomial[n-k-j,j], {j,0,(n-k)/2}];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Jun 04 2019 *)

def A073387(n,k): return sum(2^(n-k-j)*binomial(n-j,k)*binomial(n-k-j,j) for j in range(((n-k+2)//2)))
flatten([[A073387(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 03 2022

T(n, k) = 2*(p(k-1, n-k)*(n-k+1)*T(n-k+1) + q(k-1, n-k)*(n-k+2)*T(n-k))/(k!*12^k), n >= k >= 1, with T(n) = T(n, k=0) = A002605(n), else 0; p(m, n) = Sum_{j=0..m} A(m, j)*n^(m-j) and q(m, n) = Sum_{j=0..m} B(m, j)*n^(m-j) with the number triangles A(k, m) = A073403(k, m) and B(k, m) = A073404(k, m).
T(n, k) = Sum_{j=0..floor((n-k)/2)} 2^(n-k-j)*binomial(n-j, k)*binomial(n-k-j, j) if n > k, else 0.
T(n, k) = ((n-k+1)*T(n, k-1) + 2*(n+k)*T(n-1, k-1))/(6*k), n >= k >= 1, T(n, 0) = A002605(n+1), else 0.
Sum_{k=0..n} T(n, k) = A007482(n).
G.f. for column m (without leading zeros): 1/(1-2*x*(1+x))^(m+1), m>=0.
T(n,k) = 2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2, (k-n+1)/2], [-n], -2) for n>=1. - Peter Luschny, Apr 25 2016
From G. C. Greubel, Oct 03 2022: (Start)
T(n, n-1) = A005843(n), n >= 1.
T(n, n-2) = 2*A005563(n-1), n >= 2.
T(n, n-3) = 4*A159920(n-1), n >= 2.
Sum_{k=0..n} (-1)^k*T(n, k) = A001045(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A015518(n+1). (End)

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A073403 Coefficient triangle of polynomials (falling powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073404.

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A073387 Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0.

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