cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-15 of 15 results.

A025992 Expansion of 1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)).

Original entry on oeis.org

1, 22, 313, 3666, 38493, 377286, 3529681, 31947322, 282198565, 2447183310, 20920905369, 176852694018, 1481626607917, 12322682753494, 101879323774177, 838170485025354, 6867569457133749, 56077266261254238
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Bruno Berselli, May 09 2013: (Start)
a(n) - 2*a(n-1), for n>0, gives A019928 (after 1);
a(n) - 5*a(n-1), for n>0, gives A016311 (after 1);
a(n) - 7*a(n-1), for n>0, gives A016297 (after 1);
a(n) - 8*a(n-1), for n>0, gives A016296 (after 1);
a(n) - 7*a(n-1) + 10*a(n-2), for n>1, gives A016177 (after 15);
a(n) - 9*a(n-1) + 14*a(n-2), for n>1, gives A016162 (after 13);
a(n) - 10*a(n-1) + 16*a(n-2), for n>1, gives A016161 (after 12);
a(n) - 12*a(n-1) + 35*a(n-2), for n>1, gives A016131 (after 10);
a(n) - 13*a(n-1) + 40*a(n-2), for n>1, gives A016130 (after 9);
a(n) - 15*a(n-1) + 56*a(n-2), for n>1, gives A016127 (after 7);
a(n) - 20*a(n-1) +131*a(n-2) -280*a(n-3), for n>2, gives A000079 (after 4);
a(n) - 17*a(n-1) +86*a(n-2) -112*a(n-3), for n>2, gives A000351 (after 25);
a(n) - 15*a(n-1) +66*a(n-2) -80*a(n-3), for n>2, gives A000420 (after 49);
a(n) - 14*a(n-1) +59*a(n-2) -70*a(n-3), for n>2, gives A001018 (after 64),
and naturally: a(n) - 22*a(n-1) + 171*a(n-2) - 542*a(n-3) + 560*a(n-4), for n>3, gives 0 (see Harvey P. Dale in Formula lines). (End)

Links

Crossrefs

Cf. A000079, A000351, A000420, A001018, A016127, A016130, A016131, A016161, A016162, A016177, A016296, A016297, A016311, A019928.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)))); // Bruno Berselli, May 09 2013
  • Mathematica
    CoefficientList[Series[1/((1-2x)(1-5x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {22,-171,542,-560},{1,22,313,3666},30] (* Harvey P. Dale, Jan 29 2013 *)
  • PARI
    a(n) = n+=3; (5*8^n-9*7^n+5*5^n-2^n)/90 \\ Charles R Greathouse IV, Oct 03 2016
  • Python
    def A025992(n): return (5*pow(8,n+3)-9*pow(7,n+3)+pow(5,n+4)-pow(2,n+3))//90
print([A025992(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

Formula

a(0)=1, a(1)=22, a(2)=313, a(3)=3666, a(n) = 22*a(n-1) - 171*a(n-2) + 542*a(n-3) - 560*a(n-4). - Harvey P. Dale, Jan 29 2013
a(n) = (5*8^(n+3) - 9*7^(n+3) + 5^(n+4) - 2^(n+3))/90. - Yahia Kahloune, May 07 2013
E.g.f.: (1/90)*(-8*exp(2*x) + 625*exp(5*x) - 3087*exp(7*x) + 2560*exp(8*x)). - G. C. Greubel, Dec 27 2024

A036565 Triangle of numbers in which i-th row is {2^(i-j)*7^j, 0<=j<=i}; i >= 0.

Original entry on oeis.org

1, 2, 7, 4, 14, 49, 8, 28, 98, 343, 16, 56, 196, 686, 2401, 32, 112, 392, 1372, 4802, 16807, 64, 224, 784, 2744, 9604, 33614, 117649, 128, 448, 1568, 5488, 19208, 67228, 235298, 823543, 256, 896, 3136, 10976, 38416, 134456, 470596, 1647086, 5764801
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			The triangle begins as:
  1;
  2,  7;
  4, 14, 49;
  8, 28, 98, 343;
  ...

Crossrefs

Cf. A003591.
Cf. A000079 (1st column), A000420 (diagonal), A016130 (row sums).

Programs

  • Mathematica
    row[n_] := Table[SeriesCoefficient[1/((1 - 2*x)(1 - 7*x*y)), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n}]; Array[row,9,0]//Flatten (* Stefano Spezia, Aug 19 2025 *)

Formula

G.f.: 1/((1 - 2*x)(1 - 7*x*y)). - Ilya Gutkovskiy, Jun 03 2017

A096040 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^6-M)/5, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

Original entry on oeis.org

1, 7, 2, 43, 21, 3, 259, 172, 42, 4, 1555, 1295, 430, 70, 5, 9331, 9330, 3885, 860, 105, 6, 55987, 65317, 32655, 9065, 1505, 147, 7, 335923, 447896, 261268, 87080, 18130, 2408, 196, 8, 2015539, 3023307, 2015532, 783804, 195930, 32634, 3612, 252, 9
Offset: 1

Views

Author

Gary W. Adamson, Jun 17 2004

Keywords

Examples

			Triangle begins:
1;
7,       2;
43,     21,    3;
259,   172,   42,   4;
1555, 1295,  430,  70,   5;
9331, 9330, 3885, 860, 105, 6;

Crossrefs

Cf. A007318. First column gives A003464. Row sums give A016130.

Programs

  • Maple
    P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^6-M)/5 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11); # Alois P. Heinz, Oct 07 2009
  • Mathematica
    max = 11; M = Table[If[k > n, 0, Binomial[n, k]], {n, 0, max}, {k, 0, max} ];
T = (MatrixPower[M, 6] - M)/5;
Table[T[[n + 1]][[1 ;; n]] , {n, 1, max}] // Flatten (* Jean-François Alcover, May 24 2016 *)

Extensions

Edited with more terms by Alois P. Heinz, Oct 07 2009

A016201 Expansion of g.f. 1/((1-x)*(1-2*x)*(1-7*x)).

Original entry on oeis.org

1, 10, 77, 554, 3909, 27426, 192109, 1345018, 9415637, 65910482, 461375421, 3229632042, 22607432485, 158252043778, 1107764339213, 7754350440026, 54280453211253, 379963172740914, 2659742209710685, 18618195469023370, 130327368285260741, 912291578001019490, 6386041046015525037
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016130, A023000.

Programs

  • Maple
    a:=n->sum((7^(n-j+1)-2^(n-j+1))/5, j=0..n+1): seq(a(n), n=0..19); # Zerinvary Lajos, Jan 15 2007
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-2x)(1-7x)),{x,0,20}],x](* or *) LinearRecurrence[{10,-23,14},{1,10,77},20] (* Harvey P. Dale, Mar 06 2019 *)

Formula

a(n) = (49*7^n - 24*2^n + 5)/30. - Bruno Berselli, Feb 09 2011
a(0)=1, a(n) = 7*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 09 2011
From Elmo R. Oliveira, Mar 27 2025: (Start)
E.g.f.: exp(x)*(49*exp(6*x) - 24*exp(x) + 5)/30.
a(n) = 10*a(n-1) - 23*a(n-2) + 14*a(n-3).
a(n) = A016130(n+1) - A023000(n+2). (End)

Extensions

More terms from Elmo R. Oliveira, Mar 27 2025

A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 5, 13, 0, 1, 7, 25, 51, 0, 1, 9, 43, 125, 205, 0, 1, 11, 67, 259, 625, 819, 0, 1, 13, 97, 477, 1555, 3125, 3277, 0, 1, 15, 133, 803, 3355, 9331, 15625, 13107, 0, 1, 17, 175, 1261, 6505, 23517, 55987, 78125, 52429, 0, 1, 19, 223, 1875, 11605
Offset: 0

Views

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 10 2005

Keywords

Comments

Consider a 5x5 matrix M =
[n, 1, 1, 1, 1]
[1, n, 1, 1, 1]
[1, 1, n, 1, 1]
[1, 1, 1, n, 1]
[1, 1, 1, 1, n].
The n-th row of the array contains the values of the non diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non diagonal entry + (n-1)^k.)
For row r we have polynomial ((r+4)^n-(r-1)^n)/5. Corresponding g.f.s: x/((1-(r-1)x)(1-(r+4)x))
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+4.
Triangle T(n, k) = (4^(n-k-1)-(-1)^(n-k-1))/5*(binomial(k+(n-k-1),n-k-1)) gives coefficients for polynomials for the columns of the array. First four polynomial are:
1
3 + 2*k
13 + 9*k + 3*k^2
51 + 52*k + 18*k^2 + 4*k^3
...

Examples

			Array begins:
  0, 1,  3, 13,  51,  205, ...
  0, 1,  5, 25, 125,  625, ...
  0, 1,  7, 43, 259, 1555, ...
  0, 1,  9, 67, 477, 3355, ...
  0, 1, 11, 97, 803, 6505, ...
  ...

Crossrefs

Cf. A015521 (for n=0), A000351 (for n=1), A003464 (for n=2), A016130 (for n=3), A016140 (for n=4), A016153 (for n=5), A016164 (for n=6), A016174 (for n=7), A016184 (for n=8), A015441 (for n=-1), A091005 (for n=-2).

Programs

  • PARI
    MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M
for(k=0,10, for(i=0,10,print1((MM(5,k)^i)[1,2],","));print())
  • PARI
    p(n,k)=((n+4)^k-(n-1)^k)/5
for(k=0,10, for(i=0,10,print1(p(k,i),","));print())
  • PARI
    for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)),i),","));print())
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.