A073389 -id:A073389 - cp's OEIS frontend

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

A073405 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n >= 0, (generalized (2,2)-Fibonacci). Companion triangle is A073406.

Original entry on oeis.org

1, 36, 12, 1536, 888, 120, 80448, 62592, 15168, 1152, 5068800, 4813056, 1600704, 222336, 10944, 375598080, 413351424, 169917696, 32811264, 2992896, 103680, 32103751680, 39661608960, 19066503168, 4592982528

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are p(k,x) := sum(a(k,m)*x^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^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073406(k,m).

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

W. Lang, First 7 rows.

Cf. A002605, A073387, A073406, A073403.

Recursion for row polynomials defined in the comments: p(k, n)= 2*(2*(n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n+1)); q(k, n)= 4*((n+1)*p(k-1, n+1)+(n+2*(k+1))*q(k-1, n)), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

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

Original entry on oeis.org

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.

A073390 Third convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 8, 48, 240, 1080, 4512, 17856, 67776, 248880, 889600, 3109376, 10664448, 35989248, 119761920, 393676800, 1280157696, 4122985728, 13165099008, 41713192960, 131243970560, 410315433984, 1275348344832

Offset: 0

Wolfdieter Lang, Aug 02 2002

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-16,-16,56,32,-64,-64,-16).

Fourth (m=3) column of triangle A073387.

Cf. A002605, A073389.

List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+3,3)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^4 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^4, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *)

def A073390_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^4 ).list()
A073390_list(40) # G. C. Greubel, Oct 03 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073389(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, 3)*binomial(n-k, k)*2^(n-k).
a(n) = ((64 + 37*n + 5*n^2)*(n+1)*U(n+1) + 4*(11 + 7*n + n^2)*(n+2)*U(n))/(6^3), with U(n) = A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^4.

A073406 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073405.

Original entry on oeis.org

2, 36, 12, 1056, 672, 96, 43968, 40416, 10752, 864, 2396160, 2815488, 1051776, 156672, 8064, 161879040, 226492416, 105981696, 22125312, 2121984, 76032, 13044326400, 20766633984, 11446769664, 2995605504

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(k,x) := sum(a(k,m)*x^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 p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073405(k,m).

			k=2: U2(n)=2*((36+12*n)*(n+1)*U0(n+1)+(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
2; 36,12; 1056,672,96; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows.

Cf. A002605, A073387, A073403-A073405.

Recursion for row polynomials defined in the comments: p(k, n)= 2*((n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n+1)); q(k, n)= 4*((n+1)*p(k-1, n+1)+(n+2*(k+1))*q(k-1, n)), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

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

Original entry on oeis.org

2, 12, 36, 96, 672, 1056, 864, 10752, 40416, 43968, 8064, 156672, 1051776, 2815488, 2396160, 76032, 2121984, 22125312, 105981696, 226492416, 161879040, 718848, 27205632, 404656128, 2995605504

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(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!*(2^2+4*2)^k), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073403(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; 96,672,1056; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows.

Cf. A002605, A073387, A073403, A073405.

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

cp's OEIS Frontend

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

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A073405 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n >= 0, (generalized (2,2)-Fibonacci). Companion triangle is A073406.

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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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A073390 Third convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

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Magma

Mathematica

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A073406 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073405.

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

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