A073406 -id:A073406 - cp's OEIS frontend

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

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]

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.

cp's OEIS Frontend

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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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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Author

Keywords

Comments

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Crossrefs

Programs

Magma

Mathematica

SageMath

Formula