A008346 -id:A008346 - cp's OEIS frontend

A052684 Expansion of e.g.f. 1/(1-2*x^2-x^3).

1, 0, 4, 6, 96, 480, 6480, 60480, 887040, 11975040, 203212800, 3512678400, 69455232000, 1444668825600, 32953394073600, 796373690112000, 20671716409344000, 567677135241216000, 16550136029306880000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 632

Cf. A000142, A008346.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( 1/(1-2*x^2-x^3) ))); // G. C. Greubel, Jun 03 2022

spec := [S,{S=Sequence(Prod(Z,Union(Z,Z,Prod(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[1/(1-2x^2-x^3),{x,0,nn}], x] Range[ 0,nn]!] (* Harvey P. Dale, Apr 22 2012 *)
Table[n!*(Fibonacci[n]+(-1)^n), {n,0,40}] (* G. C. Greubel, Jun 03 2022 *)

[factorial(n)*(fibonacci(n) +(-1)^n) for n in (0..40)] # G. C. Greubel, Jun 03 2022

E.g.f.: 1/(1 - 2*x^2 - x^3).
D-finite recurrence: a(0)=1, a(1)=0, a(2)=4, a(n) = 2*n*(n-1)*a(n-2) + n*(n-1)*(n-2)*a(n-3).
a(n) = (n!/5)*Sum_{alpha=RootOf(-1+2*Z^2+Z^3)} (-6 + 7*alpha + 8*alpha^2)*alpha^(-1-n).
a(n) = n!*A008346(n). - R. J. Mathar, Nov 27 2011

A124377 Riordan array (1/(1-x-x^2),x/(1+x)).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 2, -1, 1, 5, 1, 3, -2, 1, 8, 4, -2, 5, -3, 1, 13, 4, 6, -7, 8, -4, 1, 21, 9, -2, 13, -15, 12, -5, 1, 34, 12, 11, -15, 28, -27, 17, -6, 1, 55, 22, 1, 26, -43, 55, -44, 23, -7, 1, 89, 33, 21

Offset: 0

Paul Barry, Oct 29 2006

First column is F(n+1). Second column is A008346. Row sums are F(n+2). Diagonal sums are A094966(n+1). Product of A007318 and A124377 is the Riordan array ((1-x)/(1-3x+x^2),x), the sequence array for F(2n+1).

			Triangle begins
1,
1, 1,
2, 0, 1,
3, 2, -1, 1,
5, 1, 3, -2, 1,
8, 4, -2, 5, -3, 1,
13, 4, 6, -7, 8, -4, 1,
21, 9, -2, 13, -15, 12, -5, 1

Cf. A000045

Number triangle T(n,k)=sum{j=0..n-k, C(j-k,n-k-j)}*[k<=n]
T(n,k)=T(n-1,k-1)+2*T(n-2,k)-T(n-2,k-1)+T(n-3,k)-T(n-3,k-1), T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014
T(n,0)=A000045(n+1), T(n,n)=1, T(n,k)=T(n-1,k-1)-T(n-1,k) for 0Philippe Deléham, Jan 12 2014

A208153 Convolution triangle based on A006053.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 4, 7, 3, 1, 9, 14, 12, 4, 1, 14, 35, 31, 18, 5, 1, 28, 70, 87, 56, 25, 6, 1, 47, 154, 207, 175, 90, 33, 7, 1, 89, 306, 504, 476, 310, 134, 42, 8, 1, 155, 633, 1137, 1274, 941, 504, 189, 52, 9, 1

Offset: 0

Philippe Deléham, Feb 24 2012

Riordan array (1/(1-x-2*x^2+x^3), x/(1-x-2*x^2+x^3)).
Subtriangle of triangle given by (0, 1, 2, -5/2, 1/10, 2/5, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Diagonal sums are A125691(n).
Row sums are A001654(n+1).
Mirror image of triangle in A188107.

			Triangle begins:
  1
  1, 1
  3, 2, 1
  4, 7, 3, 1
  9, 14, 12, 4, 1
  14, 35, 31, 18, 5, 1
Triangle (0, 1, 2, -5/2, 1/10, 2/5, 0, 0,...) DELTA (1, 0, 0, 0,...) begins:
  1
  0, 1
  0, 1, 1
  0, 3, 2, 1
  0, 4, 7, 3, 1
  0, 9, 14, 12, 4, 1
  0, 14, 35, 31, 18, 5, 1

Indranil Ghosh, Rows 1..100, flattened

Cf. A006053, A000012, A000027, A055998.

nmax=9; Flatten[CoefficientList[Series[CoefficientList[Series[1/(1 - x - 2*x^2 + x^3 - y*x), {x, 0, nmax}], x], {y, 0, nmax}], y]] (* Indranil Ghosh, Mar 10 2017 *)

T(n,k) = T(n-1,k-1) + T(n-1,k) + 2*T(n-2,k) - T(n-3,k).
G.f.: 1/(1-x-2*x^2+x^3-y*x).
Sum_{k>=0} T(n-2*k,k) = A001045(n+1).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A008346(n), A006053(n+2), A001654(n+1) for x = -1, 0, 1 respectively.

A215038 Partial sums of A066259: a(n) = Sum_{k=0..n} F(k+1)^2*F(k), n>=0, with the Fibonacci numbers F=A000045.

Original entry on oeis.org

0, 1, 5, 23, 98, 418, 1770, 7503, 31779, 134629, 570284, 2415788, 10233404, 43349461, 183631161, 777874251, 3295127934, 13958386366, 59128672790, 250473078515, 1061020985255, 4494557022121, 19039249069560, 80651553307128

Offset: 0

Wolfdieter Lang, Aug 09 2012

For a derivation of the explicit form of this sum see the link under A215308 on the partial summation formula, eq. (7).

			a(2) = 0 + 1^2*1 + 2^2*1 = 1 + 4 = 5.

Index entries for linear recurrences with constant coefficients, signature (4,3,-9,2,1).

Cf. A000045, A001655, A008346, A066258, A066259, A215308, A215037.

a(n) = Sum_{k=0..n} A066259(k) = Sum_{k=0..n} F(k+1)^2*F(k), n >= 0, with A066259(0)=0.
a(n) = (F(n+2)*F(n+1)^2 - (-1)^n*F(n) - 1)/2 = (A066258(n+1) - (-1)^n*A008346(n))/2, n >= 0.
O.g.f.: x*(1+x)/((1+x-x^2)*(1-4*x-x^2)*(1-x)) (from A066259).
E.g.f.: (2*exp(-x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2)) + exp(2*x)*(15*cosh(sqrt(15)*x) + 7*sqrt(5)*sinh(sqrt(5)*x)) - 25*exp(x))/50. - Stefano Spezia, Oct 28 2024

A230449 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A052952(n), n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 4, 1, 4, 8, 9, 8, 1, 5, 12, 17, 17, 12, 1, 6, 17, 29, 34, 29, 21, 1, 7, 23, 46, 63, 63, 50, 33, 1, 8, 30, 69, 109, 126, 113, 83, 55, 1, 9, 38, 99, 178, 235, 239, 196, 138, 88, 1, 10, 47, 137, 277, 413, 474, 435, 334, 226, 144

Offset: 0

Johannes W. Meijer, Oct 19 2013

The right hand columns of triangle T(n, k) represent the Kn2p sums of the ‘Races with Ties’ triangle A035317. See A180662 for the definitions of these sums.
The row sums lead to A094687, the convolution of Fibonacci and Jacobsthal numbers, and the alternating row sums lead to A008346.
The backwards antidiagonal sums equal Kn21(n) = (-1)^n*A175722(n).

			The first few rows of triangle T(n, k), n >= 0 and 0 <= k <= n.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1
1|  1,  1
2|  1,  2,  3
3|  1,  3,  5,   4
4|  1,  4,  8,   9,   8
5|  1,  5, 12,  17,  17,   12
6|  1,  6, 17,  29,  34,   29,   21
7|  1,  7, 23,  46,  63,   63,   50,   33
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1,  1,  3,   4,   8,   12,   21,   33
1|  1,  2,  5,   9,  17,   29,   50,   83
2|  1,  3,  8,  17,  34,   63,  113,  196
3|  1,  4, 12,  29,  63,  126,  239,  435
4|  1,  5, 17,  46, 109,  235,  474,  909
5|  1,  6, 23,  69, 178,  413,  887, 1796
6|  1,  7, 30,  99, 277,  690, 1577, 3373
7|  1,  8, 38, 137, 414, 1104, 2681, 6054

Cf. (Triangle columns) A000012, A000027, A089071, A052952, A129696

Cf. A230447, A230448

T:= proc(n, k) option remember: if k=0 then return(1) elif k=n then return(combinat[fibonacci](n+2) - (1-(-1)^n)/2) else procname(n-1,k-1)+procname(n-1,k) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End first program.
T := proc(n, k): add(A035317(k-p+n-k, k-2*p), p=0..floor(k/2)) end: A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program.

T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = F(n+2) - (1-(-1)^n)/2 = A052952(n), with F(n) = A000045(n), the Fibonacci numbers, n >= 0 and 0 <= k <= n.
T(n+p-1, n) = sum(A035317(n-k+p-1, n-2*k), k=0..floor(n/2)), n >= 0 and p >= 1.
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
Tsq(n, k) = sum(Tsq(n-1, i), i=0..k), n >= 1 and k >= 0, with Tsq(0, k) = A052952(k).
Tsq(n, k) = sum(A035317(n+k-i, k-2*i), i=0..floor(k/2)), n >= 0 and k >= 0.
Tsq(n, k) = A052952(2*n+k) - sum(A035317(n+k+i+1, k+2*i+2), i = 0..n-1)
The G.f. generates the terms in the n-th row of the square array Tsq(n, k).
G.f.: (-1)^(n)/((-1+x+x^2)*(x+1)*(x-1)^(n+1)), n >= 0.

A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 0, -1, 1, 2, 0, 0, 1, 1, 3, 1, 0, 0, -1, 1, 4, 3, 0, 0, 0, 1, 1, 5, 6, 1, 0, 0, 0, -1, 1, 6, 10, 4, 0, 0, 0, 0, 1, 1, 7, 15, 10, 1, 0, 0, 0, 0, -1, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, -1, 1, 10, 36

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

Differs from A164925 in signs.
The n-th row consists of the coefficients in the expansion of (-x)^n + (((1 + sqrt(1 + 4*x))/2)^n -((1 - sqrt(1 + 4*x))/2)^n )/sqrt(1 + 4*x).
The coefficients in the expansion of Sum_{j=0..floor((n - 1)/2)} T(n,k)*x^(n - 2*j - 1) yield the n-th row in A168561, the coefficients of the n-th Fibonacci polynomial.
Row n sums up to Fibonacci(n) + (-1)^n (A008346).

			Triangle begins:
    1;
    1, -1;
    1,  0,  1;
    1,  1,  0, -1;
    1,  2,  0,  0,  1;
    1,  3,  1,  0,  0, -1;
    1,  4,  3,  0,  0,  0, 1;
    1,  5,  6,  1,  0,  0, 0, -1;
    1,  6, 10,  4,  0,  0, 0,  0, 1;
    1,  7, 15, 10,  1,  0, 0,  0, 0, -1;
    1,  8, 21, 20,  5,  0, 0,  0, 0,  0, 1;
    1,  9, 28, 35, 15,  1, 0,  0, 0,  0, 0, -1;
    ...

Wikipedia, Fibonacci polynomials

Inspired by A123018.

Cf. A007318, A026729, A049310, A052553, A164925, A168561.

Table[Table[Binomial[n - k - 1, k], {k, 0, n}], {n, 0, 12}]//Flatten

create_list(binomial(n - k - 1, k), n, 0, 12, k, 0, n);

G.f.: 1/((1 + x*y)*(1 - y - x*y^2)).
E.g.f.: exp(-x*y) + (exp(y*(1 + sqrt(1 + 4*x))/2) - exp(y*(1 - sqrt(1 + 4*x))/2))/sqrt(1 + 4*x).
T(n,1) = A023443(n).

A099096 Riordan array (1,2-x).

Original entry on oeis.org

1, 0, 2, 0, -1, 4, 0, 0, -4, 8, 0, 0, 1, -12, 16, 0, 0, 0, 6, -32, 32, 0, 0, 0, -1, 24, -80, 64, 0, 0, 0, 0, -8, 80, -192, 128, 0, 0, 0, 0, 1, -40, 240, -448, 256, 0, 0, 0, 0, 0, 10, -160, 672, -1024, 512, 0, 0, 0, 0, 0, -1, 60, -560, 1792, -2304, 1024, 0, 0, 0, 0, 0

Offset: 0

Paul Barry, Sep 25 2004

Row sums are n+1 = Sum_{k=0..n} binomial(k,n-k)*2^(2k-n)*(-1)^(n-k). Diagonal sums are (-1)^n*A008346(n). The Riordan array (1,s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).

Triangle T(n,k), 0 <= k <= n, read by rows given by [0, -1/2, 1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008

			Rows begin
  1;
  0,   2;
  0,  -1,   4;
  0,   0,  -4,   8;
  0,   0,   1, -12,  16;
  ...

Cf. A099089.

/* Matrix power T^m formula: [T^m](n,k) = */ {T(n,k,m=1)=polcoeff((1 - (1-x +x*O(x^n))^(2^m) )^k,n)} \\ Paul D. Hanna, Nov 15 2007

Number triangle T(n, k) = binomial(k, n-k)*2^k*(-1/2)^(n-k); columns have g.f. (2x-x^2)^k.
G.f. of column k of matrix power T^m = (1 - (1-x)^(2^m))^k for k >= 0, when including the leading zeros that appear above the diagonal. - Paul D. Hanna, Nov 15 2007
T(n,k) = 2*T(n-1,k-1) - T(n-2,k-1), with T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 25 2013
G.f.: 1/(1-2*x*y+x^2*y). - R. J. Mathar, Aug 12 2015

A099494 A Chebyshev transform of Fibonacci(n)+(-1)^n.

Original entry on oeis.org

1, 0, 1, 1, -1, 0, 0, -2, 0, 1, -1, 1, 2, -1, 0, 1, -2, -1, 1, -1, 0, 2, 0, 0, 1, -1, -1, 0, -1, 0, 1, 0, 1, 1, -1, 0, 0, -2, 0, 1, -1, 1, 2, -1, 0, 1, -2, -1, 1, -1, 0, 2, 0, 0, 1, -1, -1, 0, -1, 0, 1, 0, 1, 1, -1, 0, 0, -2, 0, 1, -1, 1, 2, -1, 0, 1, -2, -1, 1, -1, 0, 2, 0, 0, 1, -1, -1

Offset: 0

Paul Barry, Oct 19 2004

A Chebyshev transform of A008346, which has g.f. 1/(1-2x^2-x^3). The image of G(x) under the Chebyshev transform is (1/(1+x^2))*G(x/(1+x^2)).

Periodic with period length 30. - Ray Chandler, Sep 08 2015

Index entries for linear recurrences with constant coefficients, signature (0,-1,1,-1,0,-1).

Cf. A000484, A008346, A014019

G.f.: (1+x^2)^2/(1+x^2-x^3+x^4+x^6).
a(n) = -a(n-2)+a(n-3)-a(n-4)-a(n-6).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(F(n-2*k)+(-1)^(n-2*k)).
a(n) = A014019(n-1) + A000484(n).

cp's OEIS Frontend

A052684 Expansion of e.g.f. 1/(1-2*x^2-x^3).

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A124377 Riordan array (1/(1-x-x^2),x/(1+x)).

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A208153 Convolution triangle based on A006053.

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A215038 Partial sums of A066259: a(n) = Sum_{k=0..n} F(k+1)^2*F(k), n>=0, with the Fibonacci numbers F=A000045.

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A230449 T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = 1 and T(n, n) = A052952(n), n >= 0 and 0 <= k <= n.

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A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.

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A099096 Riordan array (1,2-x).

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A099494 A Chebyshev transform of Fibonacci(n)+(-1)^n.

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