A000115 -id:A000115 - cp's OEIS frontend

A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

1, 9, 17, 25, 41, 49, 57, 65, 81, 105, 113, 121, 137, 145, 153, 161, 193, 201, 225, 233, 249, 257, 265, 273, 289, 329, 337, 361, 377, 385, 393, 401, 433, 441, 449, 457, 505, 513, 521, 529, 545, 553, 561, 569, 585, 609, 617, 625, 657, 713, 753, 761

Offset: 0

Clark Kimberling, Apr 10 2012

nonn

Suppose that S={-n,...,0,...,n} and that f(w,x,y,n) is a function, where w,x,y are in S. The number of ordered triples (w,x,y) satisfying f(w,x,y,n)=0, regarded as a function of n, is a sequence t of nonnegative integers. Sequences such as t/4 may also be integer sequences for all except certain initial values of n. In the following guide, such sequences are indicated in the related sequences column and may be included in the corresponding Mathematica programs.
...
sequence... f(w,x,y,n) ..... related sequences
A211415 ... w^2+x*y-1 ...... t+2, t/4, (t/4-1)/4
A211422 ... w^2+x*y ........ (t-1)/8, A120486
A211423 ... w^2+2x*y ....... (t-1)/4
A211424 ... w^2+3x*y ....... (t-1)/4
A211425 ... w^2+4x*y ....... (t-1)/4
A211426 ... 2w^2+x*y ....... (t-1)/4
A211427 ... 3w^2+x*y ....... (t-1)/4
A211428 ... 2w^2+3x*y ...... (t-1)/4
A211429 ... w^3+x*y ........ (t-1)/4
A211430 ... w^2+x+y ........ (t-1)/2
A211431 ... w^3+(x+y)^2 .... (t-1)/2
A211432 ... w^2-x^2-y^2 .... (t-1)/8
A003215 ... w+x+y .......... (t-1)/2, A045943
A202253 ... w+2x+3y ........ (t-1)/2, A143978
A211433 ... w+2x+4y ........ (t-1)/2
A211434 ... w+2x+5y ........ (t-1)/4
A211435 ... w+4x+5y ........ (t-1)/2
A211436 ... 2w+3x+4y ....... (t-1)/2
A211435 ... 2w+3x+5y ....... (t-1)/2
A211438 ... w+2x+2y ....... (t-1)/2, A118277
A001844 ... w+x+2y ......... (t-1)/4, A000217
A211439 ... w+3x+3y ........ (t-1)/2
A211440 ... 2w+3x+3y ....... (t-1)/2
A028896 ... w+x+y-1 ........ t/6, A000217
A211441 ... w+x+y-2 ........ t/3, A028387
A182074 ... w^2+x*y-n ...... t/4, A028387
A000384 ... w+x+y-n
A000217 ... w+x+y-2n
A211437 ... w*x*y-n ........ t/4, A007425
A211480 ... w+2x+3y-1
A211481 ... w+2x+3y-n
A211482 ... w*x+w*y+x*y-w*x*y
A211483 ... (n+w)^2-x-y
A182112 ... (n+w)^2-x-y-w
...
For the following sequences, S={1,...,n}, rather than
{-n,...,0,...n}. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A132188 ... w^2-x*y
A211506 ... w^2-x*y-n
A211507 ... w^2-x*y+n
A211508 ... w^2+x*y-n
A211509 ... w^2+x*y-2n
A211510 ... w^2-x*y+2n
A211511 ... w^2-2x*y ....... t/2
A211512 ... w^2-3x*y ....... t/2
A211513 ... 2w^2-x*y ....... t/2
A211514 ... 3w^2-x*y ....... t/2
A211515 ... w^3-x*y
A211516 ... w^2-x-y
A211517 ... w^3-(x+y)^2
A063468 ... w^2-x^2-y^2 .... t/2
A000217 ... w+x-y
A001399 ... w-2x-3y
A211519 ... w-2x+3y
A008810 ... w+2x-3y
A001399 ... w-2x-3y
A008642 ... w-2x-4y
A211520 ... w-2x+4y
A211521 ... w+2x-4y
A000115 ... w-2x-5y
A211522 ... w-2x+5y
A211523 ... w+2x-5y
A211524 ... w-3x-5y
A211533 ... w-3x+5y
A211523 ... w+3x-5y
A211535 ... w-4x-5y
A211536 ... w-4x+5y
A008812 ... w+4x-5y
A055998 ... w+x+y-2n
A074148 ... 2w+x+y-2n
A211538 ... 2w+2x+y-2n
A211539 ... 2w+2x-y-2n
A211540 ... 2w-3x-4y
A211541 ... 2w-3x+4y
A211542 ... 2w+3x-4y
A211543 ... 2w-3x-5y
A211544 ... 2w-3x+5y
A008812 ... 2w+3x-5y
A008805 ... w-2x-2y (repeated triangular numbers)
A001318 ... w-2x+2y
A000982 ... w+x-2y
A211534 ... w-3x-3y
A211546 ... w-3x+3y (triply repeated triangular numbers)
A211547 ... 2w-3x-3y (triply repeated squares)
A082667 ... 2w-3x+3y
A055998 ... w-x-y+2
A001399 ... w-2x-3y+1
A108579 ... w-2x-3y+n
...
Next, S={-n,...-1,1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated inequality. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.
A211545 ... w+x+y>0; recurrence degree: 4
A211612 ... w+x+y>=0
A211613 ... w+x+y>1
A211614 ... w+x+y>2
A211615 ... |w+x+y|<=1
A211616 ... |w+x+y|<=2
A211617 ... 2w+x+y>0; recurrence degree: 5
A211618 ... 2w+x+y>1
A211619 ... 2w+x+y>2
A211620 ... |2w+x+y|<=1
A211621 ... w+2x+3y>0
A211622 ... w+2x+3y>1
A211623 ... |w+2x+3y|<=1
A211624 ... w+2x+2y>0; recurrence degree: 6
A211625 ... w+3x+3y>0; recurrence degree: 8
A211626 ... w+4x+4y>0; recurrence degree: 10
A211627 ... w+5x+5y>0; recurrence degree: 12
A211628 ... 3w+x+y>0; recurrence degree: 6
A211629 ... 4w+x+y>0; recurrence degree: 7
A211630 ... 5w+x+y>0; recurrence degree: 8
A211631 ... w^2>x^2+y^2; all terms divisible by 8
A211632 ... 2w^2>x^2+y^2; all terms divisible by 8
A211633 ... w^2>2x^2+2y^2; all terms divisible by 8
...
Next, S={1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated relation.
A211634 ... w^2<=x^2+y^2
A211635 ... w^2A211790
A211636 ... w^2>=x^2+y^2
A211637 ... w^2>x^2+y^2
A211638 ... w^2+x^2+y^2A211639 ... w^2+x^2+y^2<=n
A211640 ... w^2+x^2+y^2>n
A211641 ... w^2+x^2+y^2>=n
A211642 ... w^2+x^2+y^2<2n
A211643 ... w^2+x^2+y^2<=2n
A211644 ... w^2+x^2+y^2>2n
A211645 ... w^2+x^2+y^2>=2n
A211646 ... w^2+x^2+y^2<3n
A211647 ... w^2+x^2+y^2<=3n
A063691 ... w^2+x^2+y^2=n
A211649 ... w^2+x^2+y^2=2n
A211648 ... w^2+x^2+y^2=3n
A211650 ... w^3A211790
A211651 ... w^3>x^3+y^3; see Comments at A211790
A211652 ... w^4A211790
A211653 ... w^4>x^4+y^4; see Comments at A211790

			a(1) counts these 9 triples: (-1,-1,1), (-1, 1,-1), (0, -1, 0), (0, 0, -1), (0,0,0), (0,0,1), (0,1,0), (1,-1,1), (1,1,-1).

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A120486.

t[n_] := t[n] = Flatten[Table[w^2 + x*y, {w, -n, n}, {x, -n, n}, {y, -n, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 70}] (* A211422 *)
(t - 1)/8                   (* A120486 *)

A000008 Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 19, 22, 25, 28, 31, 34, 40, 43, 49, 52, 58, 64, 70, 76, 82, 88, 98, 104, 114, 120, 130, 140, 150, 160, 170, 180, 195, 205, 220, 230, 245, 260, 275, 290, 305, 320, 341, 356, 377, 392, 413, 434, 455, 476, 497, 518, 546

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 2, 5, and 10.

There is a unique solution to this puzzle: "There are a prime number of ways that I can make change for n cents using coins of 1, 2, 5, 10 cents; but a semiprime number of ways that I can make change for n-1 cents and for n+1 cents." There is a unique solution to this related puzzle: "There are a prime number of ways that I can make change for n cents using coins of 1, 2, 5, 10 cents; but a 3-almost prime number of ways that I can make change for n-1 cents and for n+1 cents." - Jonathan Vos Post, Aug 26 2005

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 8*x^9 + 11*x^10 + ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

William Boyles, Table of n, a(n) for n = 0..10000 (terms 0...1000 from T. D. Noe)
Henry Bottomley, Initial terms of A000008, A001301, A001302, A001312, A001313
X. Gourdon and B. Salvy, Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 174
Gerhard Kirchner, Derivation of formulas
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1,-1,-1,1,0,1,-1,-1,1,0,-1,1,1,-1).

Cf. A001299, A025810.

Cf. A001300, A169718.

a000008 = p [1,2,5,10] where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 15 2013

[#RestrictedPartitions(n,{1,2,5,10}):n in [0..60]]; // Marius A. Burtea, May 07 2019

M:= Matrix(18, (i,j)-> if(i=j-1 and i<17) or (j=1 and member(i, [2,5,10,17,18])) or (i=18 and j=18) then 1 elif j=1 and member(i, [7,12,15]) then -1 else 0 fi); a:= n-> (M^(n+1))[18,1]; seq(a(n), n=0..51); # Alois P. Heinz, Jul 25 2008
# second Maple program:
a:= proc(n) local m, r; m := iquo(n, 10, 'r'); r:= r+1; ([23, 26, 35, 38, 47, 56, 65, 74, 83, 92][r]+ (3*r+ 24+ 10*m) *m) *m/6+ [1, 1, 2, 2, 3, 4, 5, 6, 7, 8][r] end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 05 2008

a[ n_] := SeriesCoefficient[ 1 / ((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)), {x, 0, n}]
a[n_, d_] := SeriesCoefficient[1/(Times@@Map[(1-x^#)&, d]), {x, 0, n}] (* general case for any set of denominations represented as a list d of coin values in cents *)
Table[Length[FrobeniusSolve[{1,2,5,10},n]],{n,0,70}] (* Harvey P. Dale, Apr 02 2012 *)
LinearRecurrence[{1, 1, -1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1}, {1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 19, 22, 25, 28}, 100] (* Vincenzo Librandi, Feb 10 2016 *)
a[ n_] := Quotient[ With[{r = Mod[n, 10, 1]}, n^3 + 27 n^2 + (191 + 3 {4, 13, 0, 5, 8, 9, 8, 5, 0, 13}[[r]]) n + 25], 600] + 1; (* Michael Somos, Mar 06 2018 *)
Table[Length@IntegerPartitions[n,All,{1,2,5,10}],{n,0,70}] (* Giorgos Kalogeropoulos, May 07 2019 *)

a(n):=floor(((n+17)*(2*n^2+20*n+81)+15*(n+1)*(-1)^n+120*((floor(n/5)+1)*((1+(-1)^mod(n,5))/2-floor(((mod(n,5))^2)/8))))/1200); /* Tani Akinari, Jun 21 2013 */

{a(n) = if( n<-17, -a(-18-n), if( n<0, 0, polcoeff( 1 / ((1 - x) * (1 - x^2) * (1 - x^5) * (1 - x^10)) + x * O(x^n), n)))}; /* Michael Somos, Apr 01 2003 */

Vec( 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)) + O(x^66) ) \\ Joerg Arndt, Oct 02 2013

{a(n) = my(r = (n-1)%10 + 1); (n^3 + 27*n^2 + (191 + 3*[4, 13, 0, 5, 8, 9, 8, 5, 0, 13][r])*n + 25)\600 + 1}; /* Michael Somos, Mar 06 2018 */

G.f.: 1 / ((1 - x) * (1 - x^2) * (1 - x^5) * (1 - x^10)). - Michael Somos, Nov 17 1999
a(n) - a(n-1) = A025810(n). - Michael Somos, Dec 15 2002
a(n) = a(n-2) + a(n-5) - a(n-7) + a(n-10) - a(n-12) - a(n-15) + a(n-17) + 1. - Michael Somos, Apr 01 2003
a(n) = -a(-18-n). - Michael Somos, Apr 01 2003
a(n) = (q+1)*(h(n) - q*(3n-10q+7)/6) with q = floor(n/10) and h(n) = A000115(n) = round((n+4)^2/20). See link "Derivation of formulas". - Gerhard Kirchner, Feb 10 2017
a(n) = floor((2*n^3 + 54*n^2 + 421*n + 15*n*(-1)^n + 24*n * ((-1)^[(n mod 5)>2] - [(n mod 5)=1]) + 1248)/1200). - Hoang Xuan Thanh, Jun 27 2025

A001304 Expansion of 1/((1-x)^2(1-x^2)(1-x^5)).

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 18, 24, 31, 39, 49, 60, 73, 87, 103, 121, 141, 163, 187, 213, 242, 273, 307, 343, 382, 424, 469, 517, 568, 622, 680, 741, 806, 874, 946, 1022, 1102, 1186, 1274, 1366, 1463, 1564, 1670, 1780, 1895, 2015, 2140, 2270, 2405, 2545, 2691, 2842

Offset: 0

N. J. A. Sloane

Ways of making change for n cents using coins of 1, 2 and 5 cents, if two different kinds of 1-cent coin are counted as different. - Matthew Vandermast, Feb 27 2003

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 113, Example (2), D(n; 1,2,4,10).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 198
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1,1,-2,0,2,-1).

First differences are in A000115.

a:= proc(n) local m, r; m:= iquo(n, 10, 'r'); r:= r+1; (53+ (135+ 100*m) *m) *m/6+ [1, 2, 4, 6, 9, 13, 18, 24, 31, 39][r]+ [0, 5, 11, 18, 26, 35, 45, 56, 68, 81][r]*m+ (r-1)*5 *m^2 end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 05 2008

CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^5)),{x,0,50}],x] (* Vincenzo Librandi, Feb 24 2012 *)
LinearRecurrence[{2,0,-2,1,1,-2,0,2,-1},{1,2,4,6,9,13,18,24,31},60] (* Harvey P. Dale, Oct 03 2018 *)

a(n)=floor((n+8)*(2*n^2+11*n+18)/120) \\ Tani Akinari, May 14 2014

G.f.: 1/((1-x)^2*(1-x^2)*(1-x^5)) = 1 / ((1+x)*(x^4+x^3+x^2+x+1)*(x-1)^4).

a(n) = floor((n+8)*(2*n^2+11*n+18)/120). - Tani Akinari, May 14 2014

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 0, 1, 2, 0, 3, 0, 5, 0, 1, 0, 1, 1, 2, 0, 1, 2, 0, 3, 0, 5, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 02 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, 1, 1,  1, ...
  0, 1, 0, 1, 0, 1, 0, 1, 0, 0,  1, 1, 0, 0,  1, ...
  0, 1, 1, 2, 0, 1, 0, 1, 1, 0,  2, 1, 1, 0,  2, ...
  0, 1, 0, 2, 1, 2, 0, 1, 1, 0,  3, 1, 0, 0,  2, ...
  0, 1, 1, 3, 0, 2, 1, 2, 1, 0,  4, 1, 2, 0,  4, ...
  0, 1, 0, 3, 0, 2, 0, 2, 1, 1,  5, 2, 0, 0,  4, ...
  0, 1, 1, 4, 1, 3, 0, 2, 2, 0,  7, 2, 2, 1,  6, ...
  0, 1, 0, 4, 0, 3, 0, 2, 1, 0,  8, 2, 0, 0,  6, ...
  0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0,  9, ...
  0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0,  9, ...
  0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.
Main diagonal gives A246721.
Cf. A246688, A246690 (the same for compositions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
      add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++ ]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A025802 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 2, 5, 3, 6, 4, 7, 5, 8, 6, 10, 7, 11, 8, 13, 10, 14, 11, 16, 13, 18, 14, 20, 16, 22, 18, 24, 20, 26, 22, 29, 24, 31, 26, 34, 29, 36, 31, 39, 34, 42, 36, 45, 39, 48, 42, 51, 45, 54, 48, 58, 51

Offset: 0

N. J. A. Sloane

a(n) is the number of partitions of n into parts 2, 4, and 5. - Hoang Xuan Thanh, Jun 18 2025

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

Cf. A000115.

CoefficientList[Series[1/((1-x^2)(1-x^4)(1-x^5)),{x,0,70}],x] (* Harvey P. Dale, Sep 15 2011 *)

From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-2) = A165190(n).
a(n)-a(n-4) = A008616(n). (End)
a(n) = floor((n^2 + n*(11+5*(-1)^n) + 53 + 27*(-1)^n)/80). - Hoang Xuan Thanh, Jun 18 2025

Showing 1-6 of 6 results.

cp's OEIS Frontend

A119537 Determinant of n X n matrices of first n^2 denumerants (A000115).

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A211422 Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.

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A000008 Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

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A001304 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^5)).

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A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Mathematica

A025802 Expansion of 1/((1-x^2)*(1-x^4)*(1-x^5)).

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Formula

A001304 Expansion of 1/((1-x)^2(1-x^2)(1-x^5)).

A025802 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)).