A001874 -id:A001874 - cp's OEIS frontend

A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

2, 7, 22, 70, 222, 705, 2238, 7105, 22556, 71608, 227332, 721705, 2291178, 7273743, 23091762, 73308814, 232731578, 738846865, 2345597854, 7446508273, 23640235416, 75050038224, 238259397096, 756395887969, 2401310279090, 7623377054503, 24201736119310

Offset: 0

Clark Kimberling, Sep 04 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
In the following guide to p-INVERT sequences using s = (1,1,0,0,0,...) = A019590, in some cases t(1,1,0,0,0,...) is a shifted version of the cited sequence:
p(S)                     t(1,1,0,0,0,...)
1 - S                       A000045 (Fibonacci numbers)
1 - S^2                     A094686
1 - S^3                     A115055
1 - S^4                     A291379
1 - S^5                     A281380
1 - S^6                     A281381
1 - 2 S                     A002605
1 - 3 S                     A125145
(1 - S)^2                   A001629
(1 - S)^3                   A001628
(1 - S)^4                   A001629
(1 - S)^5                   A001873
(1 - S)^6                   A001874
1 - S - S^2                 A123392
1 - 2 S - S^2               A291382
1 - S - 2 S^2               A124861
1 - 2 S - S^2               A291383
(1 - 2 S)^2                 A073388
(1 - 3 S)^2                 A291387
(1 - 5 S)^2                 A291389
(1 - 6 S)^2                 A291391
(1 - S)(1 - 2 S)            A291393
(1 - S)(1 - 3 S)            A291394
(1 - 2 S)(1 - 3 S)          A291395
(1 - S)(1 - 2 S)            A291393
(1 - S)(1 - 2 S)(1 - 3 S)   A291396
1 - S - S^3                 A291397
1 - S^2 - S^3               A291398
1 - S - S^2 - S^3           A186812
1 - S - S^2 - S^3 - S^4     A291399
1 - S^2 - S^4               A291400
1 - S - S^4                 A291401
1 - S^3 - S^4               A291402
1 - 2 S^2 - S^4             A291403
1 - S^2 - 2 S^4             A291404
1 - 2 S^2 - 2 S^4           A291405
1 - S^3 - S^6               A291407
(1 - S)(1 - S^2)            A291408
(1 - S^2)(1 - S)^2          A291409
1 - S - S^2 - 2 S^3         A291410
1 - 2 S - S^2 + S^3         A291411
1 - S - 2 S^2 + S^3         A291412
1 - 3 S + S^2 + S^3         A291413
1 - 2 S + S^3               A291414
1 - 3 S + S^2               A291415
1 - 4 S + S^2               A291416
1 - 4 S + 2 S^2             A291417

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 3, 2, 1)

Cf. A019590, A290890, A291000, A291219, A291728, A292479, A292480.

z = 60; s = x + x^2; p = 1 - 2 s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1]  (* A019590 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291382 *)

G.f.: (-2 - 3 x - 2 x^2 - x^3)/(-1 + 2 x + 3 x^2 + 2 x^3 + x^4).

a(n) = 2*a(n-1) + 3*a(n-2) + 2*a(n-3) + a(n-4) for n >= 5.

A377153 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 6, 27, 140, 651, 2772, 11354, 44640, 169371, 624742, 2248575, 7922124, 27397937, 93214632, 312559200, 1034507696, 3384194616, 10954244952, 35118346760, 111602517096, 351819819414, 1100912299156, 3421515852834, 10566654790176, 32441857824859, 99060134392422

Offset: 0

Seiichi Manyama, Oct 18 2024

nonn

Cf. A051286, A182884, A377145, A377148, A377152, A377158, A377159.

Cf. A001874, A089627.

a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(k, n-k)^2);

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=5, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: (Sum_{k=0..2} A089627(5,k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(11/2).

A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 6, 6, 21, 42, 21, 56, 168, 168, 56, 126, 504, 756, 504, 126, 252, 1260, 2520, 2520, 1260, 252, 462, 2772, 6930, 9240, 6930, 2772, 462, 792, 5544, 16632, 27720, 27720, 16632, 5544, 792, 1287, 10296, 36036, 72072, 90090, 72072, 36036, 10296, 1287

Offset: 0

Harlan J. Brothers, Jun 19 2010

The product of A000389 and Pascal's triangle (A007318). Level 6 of Pascal's prism (A178819) read by rows: (i+5; 5, i-j, j), i >= 0, 0 <= j <= i.

			Triangle begins:
    1;
    6,   6;
   21,  42,  21;
   56, 168, 168,  56;
  126, 504, 756, 504, 126;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Cf. A000389, A007318, A178819, A178820, A178821.

Rows sum to A054849, shallow diagonals sum to A001874.

/* As triangle */ [[Binomial(n+5,5)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // Vincenzo Librandi, Oct 23 2017

Table[Multinomial[5, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
Table[Binomial[n + 5, 5]*Binomial[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)

for(n=0,10, for(k=0,n, print1(binomial(n+5,5)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 25 2017

T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
For element a_(h, i, j) in A178819: a_(6, i, j) = (i+4; 5, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^6. - Ilya Gutkovskiy, Mar 20 2020

A238241 Riordan array (1/(1-x-x^2)^2, x/(1-x-x^2)^2).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 10, 14, 6, 1, 20, 40, 27, 8, 1, 38, 105, 98, 44, 10, 1, 71, 256, 315, 192, 65, 12, 1, 130, 594, 924, 726, 330, 90, 14, 1, 235, 1324, 2534, 2472, 1430, 520, 119, 16, 1, 420, 2860, 6588, 7776, 5522, 2535, 770, 152, 18, 1, 744, 6020, 16407, 22968

Offset: 0

Philippe Deléham, Feb 20 2014

Row sums are A097472(n).

			Triangle begins:
1;
2, 1;
5, 4, 1;
10, 14, 6, 1;
20, 40, 27, 8, 1;
38, 105, 98, 44, 10, 1;
71, 256, 315, 192, 65, 12, 1;
130, 594, 924, 726, 330, 90, 14, 1;
...

Cf. A037027, A152440
Cf. Diagonals: A000012, A005843, A014106
Cf. Columns: A001629, A001872, A001874

T[0, 0] = 1; T[n_, k_] := SeriesCoefficient[-1/(x*y - x^4 - 2*x^3 + x^2 + 2*x - 1), {x, 0, n}, {y, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Vladimir Kruchinin *)

T(n,k) = A037027(n+k+1, 2*k+1).
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k) - 2*T(n-3,k) - T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: -1/(x*y-x^4-2*x^3+x^2+2*x-1). - Vladimir Kruchinin, Apr 29 2015

A036684 T(n+5,5) with T as in A036355.

Original entry on oeis.org

8, 38, 149, 478, 1390, 3736, 9496, 23080, 54127, 123230, 273653, 594878, 1269532, 2665912, 5518900, 11280856, 22797331, 45599656, 90362560, 177550600, 346157050, 670060100, 1288497590, 2462607020, 4679908400, 8846662634

Offset: 0

Floor van Lamoen

Index entries for linear recurrences with constant coefficients, signature (6,-9,-10,30,6,-41,-6,30,10,-9,-6,-1).

G.f.: (3x^4+6x^3-7x^2-10x+8)/(1-x-x^2)^6.

a(n)=3*A001872(n)+4*A001873(n)+A001874(n). - R. J. Mathar, Jul 03 2022

cp's OEIS Frontend

A291382 p-INVERT of (1,1,0,0,0,0,...), where p(S) = 1 - 2 S - S^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A377153 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A238241 Riordan array (1/(1-x-x^2)^2, x/(1-x-x^2)^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A036684 T(n+5,5) with T as in A036355.

Views

Author

Keywords

Links

Formula