A058403 -id:A058403 - cp's OEIS frontend

A058402 Coefficient triangle of polynomials (rising powers) related to Pell number convolutions. Companion triangle is A058403.

1, 22, 8, 588, 376, 56, 19656, 17024, 4576, 384, 801360, 848096, 313504, 48256, 2624, 38797920, 47494272, 21685888, 4643072, 468608, 17920, 2181332160, 2986217856, 1590913920, 424509952, 60136448, 4307456, 122368, 139864717440

Offset: 0

Wolfdieter Lang, Dec 11 2000

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 P0(n) := A000129(n+1) n >= 0, (Pell numbers starting with P0(0)=1) with itself is Pk( n) := A054456(n+k,k) = (p(k-1,n)*(n+1)*2*P0(n+1) + q(k-1,n)*(n+2)*P0(n))/(k!*8^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)= A058403(k,m).

			k=2: P2(n)=((22+8*n)*(n+1)*2*P0(n+1)+(20+8*n)*(n+2)*P0(n))/128, cf. A054457.
1; 22,8; 588,376,56; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows, also for A058403.

Cf. A000129, A054456, A058403, A058404-5 (falling powers).

Recursion for row polynomials defined in the comments: p(k, n)= 4*(n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n); q(k, n)= 4*(n+1)*p(k-1, n+1)+2*(n+2*(k+1))*q(k-1, n+1), k >= 1. [Corrected by Sean A. Irvine, Aug 05 2022]

Link and cross-references added by Wolfdieter Lang, Jul 31 2002

A073385 Eighth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

Original entry on oeis.org

1, 18, 189, 1500, 9945, 58014, 307197, 1507176, 6950295, 30443270, 127666539, 515754252, 2017069431, 7667214570, 28419251715, 102997948704, 365832349542, 1275914693196, 4376992440590

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n) with U(n) = A000129(n+1) see the row polynomials of triangles A058402 and A058403 and the comment there.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-135,528,-1044,504,1764,-2448,-1422,3308,1422, -2448,-1764,504,1044,528,135,18,1).

Ninth (m=8) column of triangle A054456.

Cf. A000129, A058402, A058403, A073384.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^9 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-(2+x)x)^9,{x,0,20}],x] (* Harvey P. Dale, Apr 26 2017 *)

taylor( 1/(1-2*x-x^2)^9, x, 0,27).list() # G. C. Greubel, Oct 03 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073384(k).
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k)*binomial(n-k+8, 8)*binomial(n-k, k).
G.f.: 1/(1-(2+x)*x)^9.
a(n) = F''''''''(n+9, 2)/8!, that is, 1/8! times the 8th derivative of the (n+9)-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

A073386 Ninth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

Original entry on oeis.org

1, 20, 230, 1980, 14135, 88264, 497860, 2591160, 12630475, 58295380, 256887774, 1087825180, 4449607565, 17654254880, 68177369040, 257006941664, 948023601910, 3428968838680, 12182953719860

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n) with U(n) = A000129(n+1) see the row polynomials of triangles A058402 and A058403 and the comment there.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (20,-170,780,-1965,2064,1800,-6480,1710,8600,-3772, -8600,1710,6480,1800,-2064,-1965,-780,-170,-20,-1).

Tenth (m=9) column of triangle A054456.

Cf. A000129, A058402, A058403, A073385.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^10 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-2*x-x^2)^10, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *)
LinearRecurrence[{20,-170,780,-1965,2064,1800,-6480,1710,8600,-3772,-8600,1710,6480,1800,-2064,-1965,-780,-170,-20,-1},{1,20,230,1980,14135,88264,497860,2591160,12630475,58295380,256887774,1087825180,4449607565,17654254880,68177369040,257006941664,948023601910,3428968838680,12182953719860,42585118702280},20] (* Harvey P. Dale, Nov 20 2022 *)

def A073386_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^10 ).list()
A073386_list(40) # G. C. Greubel, Oct 03 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073385(k).
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k)*binomial(n-k+9, 9)*binomial(n-k, k).
G.f.: 1/(1-(2+x)*x)^10.
a(n) = F'''''''''(n+10, 2)/9!, that is, 1/9! times the 9th derivative of the (n+10)th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

cp's OEIS Frontend

A058402 Coefficient triangle of polynomials (rising powers) related to Pell number convolutions. Companion triangle is A058403.

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A073385 Eighth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

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A073386 Ninth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

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Author

Keywords

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Programs

Magma

Mathematica

SageMath

Formula