A054457 -id:A054457 - cp's OEIS frontend

A054456 Convolution triangle of A000129(n) (Pell numbers).

1, 2, 1, 5, 4, 1, 12, 14, 6, 1, 29, 44, 27, 8, 1, 70, 131, 104, 44, 10, 1, 169, 376, 366, 200, 65, 12, 1, 408, 1052, 1212, 810, 340, 90, 14, 1, 985, 2888, 3842, 3032, 1555, 532, 119, 16, 1, 2378, 7813, 11784, 10716, 6482, 2709, 784, 152, 18, 1, 5741, 20892, 35223

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) (increasing powers of x) is Pell(z)/(1-x*z*Pell(z)) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0).
Column sequences are A000129(n+1), A006645(n+1), A054457(n) for m=0..2.
Riordan array (1/(1-2x-x^2),x/(1-2x-x^2)). - Paul Barry, Mar 15 2005
As a Riordan array, this factors as (1/(1-x^2),x/(1-x^2))*(1/(1-2x),x/(1-2x)), [abs(A049310) times square of A007318, or A038207]. - Paul Barry, Jul 28 2005
Coefficients of polynomials defined by P(x, 0) = 1; P(x, 1) = 2 - x; P(x, n) = (2 - x)*P(x, n - 1) + P(x, n - 2). - Roger L. Bagula, Mar 24 2008
Subtriangle (obtained by dropping the first column) of the triangle given by (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013
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 of the letter 0. - Milan Janjic, Jan 14 2017

			Fourth row polynomial (n=3): p(3,x)= 12+14*x+6*x^2+x^3
Triangle begins:
{1},
{2, 1},
{5, 4, 1},
{12, 14, 6, 1},
{29, 44, 27, 8, 1},
{70, 131,104, 44, 10, 1},
{169, 376, 366, 200, 65, 12, 1},
{408, 1052, 1212, 810, 340, 90, 14, 1},
{985, 2888, 3842, 3032, 1555, 532, 119, 16, 1},
{2378, 7813, 11784, 10716, 6482, 2709, 784, 152, 18, 1},
{5741, 20892, 35223, 36248, 25235, 12432, 4396, 1104, 189, 20, 1},
The triangle (0, 2, 1/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 2, 1
0, 5, 4, 1
0, 12, 14, 6, 1
0, 29, 44, 27, 8, 1 - _Philippe Deléham_, Feb 19 2013

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A000129. Row sums: A006190(n+1).

Cf. A129844.

G := 1/(1-(x+2)*z-z^2): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], x, j), j = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Aug 30 2015
T := (n,k) -> `if`(n=0,1,2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n], -1)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # Peter Luschny, Apr 25 2016
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, A000129); # Peter Luschny, Oct 19 2022

P[x_, 0] := 1; P[x_, 1] := 2 - x; P[x_, n_] := P[x, n] = (2 - x) P[x, n - 1] + P[x, n - 2]; Table[Abs@ CoefficientList[P[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Mar 24 2008, edited by Michael De Vlieger, Apr 25 2018 *)

a(n, m) := ((n-m+1)*a(n, m-1) + (n+m)*a(n-1, m-1))/(4*m), n >= m >= 1, a(n, 0)= P(n+1)= A000129(n+1) (Pell numbers without P(0)), a(n, m) := 0 if n

G.f. for column m: Pell(x)*(x*Pell(x))^m, m >= 0, with Pell(x) G.f. for A000129(n+1).

Number triangle T(n, k) with T(n, 0)=A000129(n), T(1, 1)=1, T(n, k)=0 if k>n, T(n, k)=T(n-1, k-1)+T(n-2, k)+2T(n-1, k) otherwise; T(n, k)=if(k<=n, sum{j=0..floor((n-k)/2), C(n-j, k)C(n-k-j, j)2^(n-2j-k)}; - Paul Barry, Mar 15 2005

Bivariate g.f. G(x,z) = 1/[1 - (2 + x)z - z^2]. G.f. for column k = z^k/(1 - 2z - z^2)^{k+1} (k>=0). - Emeric Deutsch, Aug 30 2015

T(n,k) = 2^(n-k)*C(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n],-1)) for n>=1. - Peter Luschny, Apr 25 2016

A006645 Self-convolution of Pell numbers (A000129).

Original entry on oeis.org

0, 0, 1, 4, 14, 44, 131, 376, 1052, 2888, 7813, 20892, 55338, 145428, 379655, 985520, 2545720, 6547792, 16777993, 42847988, 109099078, 277040572, 701794187, 1773851304, 4474555476, 11266301976, 28318897549, 71070913036, 178106093666, 445740656420, 1114147888655

Offset: 0

N. J. A. Sloane

			G.f. = x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + ...

R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (The sequences w_n and z_n)

Michael De Vlieger, Table of n, a(n) for n = 0..2605
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

a(n)= A054456(n-1, 1), n>=1 (second column of triangle), A054457.

a:= n-> (Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [4, -2, -4, -1][i] else 0 fi)^n) [1,3]: seq(a(n), n=0..40); # Alois P. Heinz, Oct 28 2008

pell[n_] := Simplify[ ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2])]; a[n_] := First[ ListConvolve[ pp = Array[pell, n+1, 0], pp]]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 21 2011 *)
Table[(n Fibonacci[n - 1, 2] + (n - 1) Fibonacci[n, 2])/4, {n, 0, 30}] (* Vladimir Reshetnikov, May 08 2016 *)

taylor( mul(x/(1 - 2*x - x^2) for i in range(1,3)),x,0,28) # Zerinvary Lajos, Jun 03 2009

a(n) = Sum_{k=0..n} b(k)*b(n-k) with b(k) := A000129(k).
a(n) = Sum_{k=0..floor((n-2)/2)} 2^(n-2)*(n-k-1)*binomial(n-2-k, k)*(1/4)^k, n >= 2.
From Wolfdieter Lang, Apr 11 2000: (Start)
a(n) = ((n-1)*P(n) + n*P(n-1))/4, P(n)=A000129(n).
G.f.: (x/(1 - 2*x - x^2))^2. (End)
a(n) = F'(n, 2), the derivative of the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

Sum formulas and cross-references added by Wolfdieter Lang, Aug 07 2002

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

Original entry on oeis.org

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

A058404 Coefficient triangle of polynomials (falling powers) related to Pell number convolutions. Companion triangle is A058405.

Original entry on oeis.org

1, 8, 22, 56, 376, 588, 384, 4576, 17024, 19656, 2624, 48256, 313504, 848096, 801360, 17920, 468608, 4643072, 21685888, 47494272, 38797920, 122368, 4307456, 60136448, 424509952, 1590913920, 2986217856, 2181332160, 835584, 38055936

Offset: 0

Wolfdieter Lang, Dec 11 2000

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 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^(k-m),m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A058405(k,m).
a(k,0)= A057084(k), k >= 0 (conjecture).

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

W. Lang, First 7 rows, also for A058405.

Cf. A000129, A054456, A058405, A054457, A057084, A058402-3 (rising powers).

Recursion for row polynomials defined in the comments: see A058402.

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

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

Original entry on oeis.org

2, 20, 8, 360, 288, 48, 9840, 11360, 3520, 320, 363360, 522752, 225344, 37888, 2176, 16776000, 27849600, 14871296, 3491072, 373504, 14848, 922158720, 1692808704, 1053556480, 308703232, 46459904, 3467264, 101376, 58499239680, 115821927936

Offset: 0

Wolfdieter Lang, Dec 11 2000

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 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 p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A058402(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.
2; 20,8; 360,288,48; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows, also for A058402.

Cf. A000129, A054456, A058402, A058404-5 (falling powers).

Recursion for row polynomials defined in the comments: see A058402.

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

A073380 Third convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

Original entry on oeis.org

1, 8, 44, 200, 810, 3032, 10716, 36248, 118435, 376240, 1167720, 3553840, 10636180, 31375440, 91392040, 263266512, 750922021, 2123059448, 5955034740, 16584106040, 45884989054, 126202397032

Offset: 0

Wolfdieter Lang, Aug 02 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8
Index entries for linear recurrences with constant coefficients, signature (8,-20,8,26,-8,-20,-8,-1).

Fourth (m=3) column of triangle A054456, A054457 (m=2).

Cf. A000129.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^4 )); // G. C. Greubel, Oct 02 2022

CoefficientList[Series[1/(1-2*x-x^2)^4, {x,0,40}], x] (* G. C. Greubel, Oct 02 2022 *)
LinearRecurrence[{8,-20,8,26,-8,-20,-8,-1},{1,8,44,200,810,3032,10716,36248},30] (* Harvey P. Dale, Feb 18 2023 *)

def A073380_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^4 ).list()
A073380_list(30) # G. C. Greubel, Oct 02 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A054457(k).
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+3, 3) * binomial(n-k, k).
a(n) = ((147 +94*n +14*n^2)*(n+1)*U(n+1) + 3*(15 +12*n +2*n^2)*(n+2)*U(n))/ (3*2^7), with U(n) = A000129(n+1), n >= 0.
G.f.: 1/(1-(2+x)*x)^4.
a(n) = F'''(n+4, 2)/6, that is, 1/6 times the 3rd derivative of the (n+4)th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

A058405 Coefficient triangle of polynomials (falling powers) related to Pell number convolutions. Companion triangle is A058404.

Original entry on oeis.org

2, 8, 20, 48, 288, 360, 320, 3520, 11360, 9840, 2176, 37888, 225344, 522752, 363360, 14848, 373504, 3491072, 14871296, 27849600, 16776000, 101376, 3467264, 46459904, 308703232, 1053556480, 1692808704, 922158720, 692224, 30834688

Offset: 0

Wolfdieter Lang, Dec 11 2000

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 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 p(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A058404(k,m).

			k=2: P2(n)=((8*n+22)*(n+1)*2*P0(n+1)+(8*n+20)*(n+2)*P0(n))/128, cf. A054457.
2; 8,20; 48,288,360; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0)

W. Lang, First 7 rows, also for A058404.

Cf. A000129, A054456, A058404, A054457, A058402-3 (rising powers).

Recursion for row polynomials defined in the comments: see A058402.

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

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A054456 Convolution triangle of A000129(n) (Pell numbers).

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A006645 Self-convolution of Pell numbers (A000129).

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

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A058404 Coefficient triangle of polynomials (falling powers) related to Pell number convolutions. Companion triangle is A058405.

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

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A073380 Third convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

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A058405 Coefficient triangle of polynomials (falling powers) related to Pell number convolutions. Companion triangle is A058404.

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