A073401 -id:A073401 - cp's OEIS frontend

A073402 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073401.

2, 33, 9, 831, 396, 45, 28236, 18297, 3744, 243, 1210140, 968679, 273483, 32481, 1377, 62686440, 58920534, 20681811, 3418767, 268029, 8019, 3810867480, 4075425738, 1683064737, 347584284, 38186478, 2130138, 47385

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) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^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)= A073401(k,m).

			k=2: U2(n)=((30+9*n)*(n+1)*U0(n+1)+(33+9*n)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 33,9; 831,396,45; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073399, A073372, A073400.

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

A073399 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n>=0, (generalized (1,2)-Fibonacci). Companion triangle is A073400.

Original entry on oeis.org

1, 9, 30, 63, 531, 1050, 405, 6165, 29610, 44520, 2511, 59454, 502821, 1789614, 2245320, 15309, 517104, 6686631, 41182344, 120133692, 131891760, 92583, 4214349, 76790673, 714174327, 3559509360, 8966770308, 8862693840

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) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^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)= A073400(k,m).

			k=2: U2(n)=((9*n+30)*(n+1)*U0(n+1)+(9*n+33)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 9,30; 63,531,1050; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073400-A073402, A073372.

Recursion for row polynomials defined in the comments: see A073401.

A073400 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073399.

Original entry on oeis.org

2, 9, 33, 45, 396, 831, 243, 3744, 18297, 28236, 1377, 32481, 273483, 968679, 1210140, 8019, 268029, 3418767, 20681811, 58920534, 62686440, 47385, 2130138, 38186478, 347584284, 1683064737, 4075425738, 3810867480

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) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^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)= A073399(k,m).

			k=2: U2(n)=((9*n+30)*(n+1)*U0(n+1)+(9*n+33)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 9,33; 45,396,831; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073399, A073401.

Recursion for row polynomials defined in the comments: see A073401.

A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 9, 63, 345, 1665, 7227, 29073, 109791, 394020, 1354210, 4486482, 14397318, 44932446, 136817370, 407566350, 1190446866, 3415935699, 9645169743, 26836557825, 73670997015, 199751003991, 535449185469

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n) with U(n) = A001045(n+1) see A073370 and the row polynomials of triangles A073399 and A073400.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-18,-60,234,126,-1176,36,3519,-479,-7038,144, 9408,2016,-7488,-3840,2304,2304,512).

Ninth (m=8) column of triangle A073370.

Cf. A001045, A073377, A073399, A073400, A073401.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^9 )); // G. C. Greubel, Oct 01 2022

CoefficientList[Series[1/((1+x)*(1-2*x))^9, {x,0,40}], x] (* G. C. Greubel, Oct 01 2022 *)

def A073378_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*(1-2*x))^9 ).list()
A073378_list(40) # G. C. Greubel, Oct 01 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073377(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8) * binomial(n-k, k) * 2^k.
G.f.: 1/(1-(1+2*x)*x)^9 = 1/((1+x)*(1-2*x))^9.

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A073402 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073401.

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A073399 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n>=0, (generalized (1,2)-Fibonacci). Companion triangle is A073400.

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A073400 Coefficient triangle of polynomials (falling powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073399.

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A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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