A073405 -id:A073405 - cp's OEIS frontend

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

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]

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.

A073397 Eighth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 18, 198, 1680, 12060, 76824, 446952, 2420352, 12363120, 60151520, 280833696, 1265442048, 5528697408, 23507763840, 97575960960, 396398370816, 1579498956288, 6184543546368, 23833455191040, 90522348871680, 339263015528448, 1255995653197824, 4597442198728704

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n), with U(n) = A002605(n), see A073387 and the row polynomials of triangles A073405 and A073406.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-126,384,-144,-2016,3360,4608,-12384,-8512, 24768,18432,-26880,-32256,4608,24576,16128,4608,512).

Ninth (m=8) column of triangle A073387.

Cf. A002605, A073387, A073394, A073405, A073406.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^9 )); // G. C. Greubel, Oct 06 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^9, {x,0,30}], x] (* G. C. Greubel, Oct 06 2022 *)

def A073397_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^9 ).list()
A073397_list(30) # G. C. Greubel, Oct 06 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073394(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8)*binomial(n-k, k)*2^(n-k).
G.f.: 1/(1-2*x*(1+x))^9.

A073398 Ninth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 20, 240, 2200, 16940, 115104, 711040, 4072640, 21930480, 112157760, 549010176, 2587777920, 11802273600, 52287866880, 225756241920, 952486588416, 3935984616960, 15961485957120, 63628396339200, 249702113464320, 965924035135488, 3687247950397440

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n), with U(n) = A002605(n), see A073387 and the row polynomials of triangles A073405 and A073406.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-160,600,-660,-2496,7680,1920,-28320,7040, 66560,-14080,-113280,-15360,122880,79872,-42240,-76800,-40960,-10240,-1024).

Tenth (m=9) column of triangle A073387.

Cf. A002605, A073397, A073405, A073406.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^10 )); // G. C. Greubel, Oct 06 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^10, {x,0,30}], x] (* G. C. Greubel, Oct 06 2022 *)

def A073398_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^10 ).list()
A073398_list(30) # G. C. Greubel, Oct 06 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073397(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+9, 9)*binomial(n-k, k)*2^(n-k).
G.f.: 1/(1-2*x*(1+x))^10.

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

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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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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A073397 Eighth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

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A073398 Ninth convolution of A002605(n) (generalized (2,2)-Fibonacci), n>=0, with itself.

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Magma

Mathematica

SageMath

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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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Examples

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