A054456 -id:A054456 - cp's OEIS frontend

A073383 Sixth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

1, 14, 119, 784, 4396, 22008, 101220, 435696, 1777986, 6943244, 26129950, 95282992, 338108876, 1171554776, 3975215844, 13239402960, 43364985867, 139925413866, 445409413421, 1400429394784, 4353771487912

Offset: 0

Wolfdieter Lang, Aug 02 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-77,196,-161,-238,427,184,-427,-238,161,196,77, 14,1).

Seventh (m=6) column of triangle A054456, A073382.

Cf. A000129.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^7 )); // G. C. Greubel, Oct 02 2022

CoefficientList[Series[1/(1-2*x-x^2)^7, {x,0,70}], x] (* G. C. Greubel, Oct 02 2022 *)
LinearRecurrence[{14,-77,196,-161,-238,427,184,-427,-238,161,196,77,14,1},{1,14,119,784,4396,22008,101220,435696,1777986,6943244,26129950,95282992,338108876,1171554776},30] (* Harvey P. Dale, Apr 26 2025 *)

def A073383_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^7 ).list()
A073383_list(40) # 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) = A073382(k).
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+6, 6) * binomial(n-k, k).
a(n) = (7*(173205 +212028*n +96812*n^2 +20728*n^3 +2092*n^4 +80*n^5)*(n+1)* U(n+1) + (262125 +435150*n +232364*n^2 +54548*n^3 +5836*n^4 +232*n^5)*(n+2)* U(n) )/(6!*8^4), with U(n) = A000129(n+1), n >= 0.
G.f.: 1/(1-(2+x)*x)^7.
a(n) = F''''''(n+7, 2)/6!, that is, 1/6! times the 6th derivative of the (n+7)-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

A073384 Seventh convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

Original entry on oeis.org

1, 16, 152, 1104, 6756, 36624, 181224, 834768, 3628746, 15035504, 59829704, 229977904, 857894388, 3117321456, 11067753144, 38492230704, 131417200419, 441252045408, 1459330704656, 4760342849504

Offset: 0

Wolfdieter Lang, Aug 02 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-104,336,-476,-112,1064,-432,-1222,432,1064, 112,-476,-336,-104,-16,-1).

Eighth (m=7) column of triangle A054456, A073383.

Cf. A000129.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^8 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-2*x-x^2)^8, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *)

def A073384_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^8 ).list()
A073384_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) = A073383(k).
a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+7, 7) * binomial(n-k, k).
a(n) = ((34083315 +46659654*n +24858030*n^2 +6632968*n^3 +939632*n^4 +67304*n^5 + 1912*n^6)*(n+1)*U(n+1) + (7204365 +13225068*n +8230910*n^2 +2411744*n^3 + 362968*n^4 +27088*n^5 +792*n^6)*(n+2)*U(n))/(2^18*3^2*5*7), with U(n) = A000129(n+1), n >= 0.
G.f.: 1/(1-(2+x)*x)^8.
a(n) = F'''''''(n+8, 2)/7!, that is, 1/7! times the 7th derivative of the (n+8)-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

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

A112899 A skew Pell-Pascal triangle.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 0, 4, 12, 0, 0, 1, 14, 29, 0, 0, 0, 6, 44, 70, 0, 0, 0, 1, 27, 131, 169, 0, 0, 0, 0, 8, 104, 376, 408, 0, 0, 0, 0, 1, 44, 366, 1052, 985, 0, 0, 0, 0, 0, 10, 200, 1212, 2888, 2378, 0, 0, 0, 0, 0, 1, 65, 810, 3842, 7813, 5741, 0, 0, 0, 0, 0, 0, 12, 340, 3032, 11784

Offset: 0

Paul Barry, Oct 05 2005

Main diagonal is A000129. Row sums are A002605. Column sums are A006190(n+1).
A skewed version of the Riordan array (1/(1-2x-x^2), x/(1-2x-x^2)), see A054456. - Philippe Deléham, Nov 21 2007
Triangle, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,...] DELTA [2,1/2,-1/2,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 30 2010

			Rows begin:
  1;
  0,   2;
  0,   1,   5;
  0,   0,   4,  12;
  0,   0,   1,  14,  29;
  0,   0,   0,   6,  44,  70;
  0,   0,   0,   1,  27, 131, 169;
  0,   0,   0,   0,   8, 104, 376, 408;

Cf. A111006, A112906. - Philippe Deléham, Jan 30 2010

G.f.: 1/(1-2*x*y*(1+x/2)-x^2*y^2).
T(n, k) = Sum_{j=0..floor((2*k-n)/2)} C(k-j, n-k)*C(2*k-n-j, j)*2^(2*k-2*j-n). [corrected by Jason Yuen, Jan 21 2025]
T(n, k) = 2*T(n-1, k-1) + T(n-2, k-1) + T(n-2, k-2).

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

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

A132964 Convolution triangle of A006190.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 109, 126, 57, 12, 1, 360, 516, 306, 94, 15, 1, 1189, 2034, 1491, 600, 140, 18, 1, 3927, 7807, 6813, 3385, 1035, 195, 21, 1, 12970, 29382, 29737, 17568, 6630, 1638, 259, 24, 1, 42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1

Offset: 0

Philippe Deléham, Nov 24 2007

As a Riordan array, this is (1/(1-3x-x^2),x/(1-3x-x^2)).

T(n,k) is the number of words of length n over {0,1,2,3,4} having k letters 4 and avoiding runs of odd length for the letter 0. - Milan Janjic, Jan 14 2017

			Triangle begins:
      1;
      3,      1;
     10,      6,      1;
     33,     29,      9,     1;
    109,    126,     57,    12,     1;
    360,    516,    306,    94,    15,     1;
   1189,   2034,   1491,   600,   140,    18,    1;
   3927,   7807,   6813,  3385,  1035,   195,   21,   1;
  12970,  29382,  29737, 17568,  6630,  1638,  259,  24,  1;
  42837, 108923, 125406, 85826, 38493, 11739, 2436, 332, 27, 1;
  ...

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

Cf. A006190, A037027, A054456, A112906.

Sum_{k=0..n} T(n,k) = A001076(n+1).
Sum_{k=0..floor(n/2)} T(n-k,k) = A007482(n).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) + T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or k>n. - Philippe Deléham, Dec 08 2013

cp's OEIS Frontend

A073383 Sixth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A073384 Seventh convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A112899 A skew Pell-Pascal triangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A132964 Convolution triangle of A006190.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula