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 11-19 of 19 results.

A111837 Number of partitions of 8^n into powers of 8, also equals the row sums of triangle A111835, which shifts columns left and up under matrix 8th power.

Original entry on oeis.org

1, 2, 10, 298, 53674, 58573738, 409251498922, 19046062579215274, 6071277235712979102634, 13531779463193107731083553706, 214224474679766323250278564215516074, 24390479071277895100812271376578637910371242, 20173309182842708837666031701435147789403500172143530
Offset: 0

Views

Author

Gottfried Helms and Paul D. Hanna, Aug 22 2005

Keywords

Links

Crossrefs

Cf. A111835, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111827 (q=6), A111832 (q=7). Column 8 of A145515.

Programs

  • PARI
    a(n,q=8)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i || j==1,B[i,j]=1,B[i,j]=(A^q)[i-1,j-1]);));A=B); return(sum(k=0,n,A[n+1,k+1])))

Formula

a(n) = [x^(8^n)] 1/Product_{j>=0} (1-x^(8^j)).

A145513 Number of partitions of 10^n into powers of 10.

Original entry on oeis.org

1, 2, 12, 562, 195812, 515009562, 10837901390812, 1899421190329234562, 2851206628197445401265812, 37421114946843687272702534859562, 4362395890943439751990308572939648140812, 4573514084633441973328831327010967245403925484562, 43557001521047571730475817291330175020887917015964570015812
Offset: 0

Views

Author

Alois P. Heinz, Oct 11 2008

Keywords

Comments

a(n) = A179051(10^n); for n>0: a(n) = A179052(10^(n-1)). - Reinhard Zumkeller, Jun 27 2010

Examples

			a(1) = 2, because there are 2 partitions of 10^1 into powers of 10: [1,1,1,1,1,1,1,1,1,1], [10].

Links

Crossrefs

Cf. 10th column of A145515, A007318.
Cf. A011557, A002577, A078125.

Programs

  • Haskell
    import Data.MemoCombinators (memo2, list, integral)
a145513 n = a145513_list !! n
a145513_list = f [1] where
   f xs = (p' xs $ last xs) : f (1 : map (* 10) xs)
   p' = memo2 (list integral) integral p
   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, Nov 27 2015
  • Maple
    g:= proc(b,n,k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add(g(b-t, n, k) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1,n,10): seq(a(n), n=0..13);
  • Mathematica
    g[b_, n_, k_] := g[b, n, k] = Module[{t}, Which[b < 0, 0, b == 0 || n == 0 || k <= 1, 1, b >= n, Sum[g[b - t, n, k]*Binomial[n + 1, t] *(-1)^(t + 1), {t, 1, n + 1}], True, g[b - 1, n, k] + g[b*k, n - 1, k]]]; a[n_] := g[1, n, 10]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 01 2017, after Alois P. Heinz *)

Formula

a(n) = [x^(10^n)] 1/Product_{j>=0} (1-x^(10^j)).

A125802 Column 4 of table A125800; also equals row sums of matrix power A078122^4.

Original entry on oeis.org

1, 5, 35, 485, 15200, 1144664, 215155493, 103674882878, 130648799730635, 437302448840089232, 3936208033244539574405, 96244898501021613327012635, 6446494058446469307795159512465, 1191218783863555524342034469450207222
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.

Crossrefs

Cf. A125800, A078122; other columns: A078125, A078124, A125801, A125803.

Programs

  • PARI
    a(n)=local(p=4,q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125803 Column 5 of table A125800; also equals row sums of matrix power A078122^5.

Original entry on oeis.org

1, 6, 51, 861, 32856, 3013980, 690729981, 406279238154, 625750288074015, 2563196032703643450, 28270494794022487841733, 848050124165724284639262951, 69769378541879435090796205851249
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.

Crossrefs

Cf. A125800, A078122; other columns: A078125, A078124, A125801, A125802.

Programs

  • PARI
    a(n)=local(p=5,q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A078123 Square of infinite lower triangular matrix A078122.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 23, 51, 18, 1, 239, 861, 477, 54, 1, 5828, 32856, 25263, 4347, 162, 1, 342383, 3013980, 3016107, 699813, 39285, 486, 1, 50110484, 690729981, 865184724, 253656252, 19053063, 354051, 1458, 1, 18757984046, 406279238154
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2002

Keywords

Examples

			Square of A078122 = A078123 as can be seen by 4 X 4 submatrix:
[1,_0,_0,0]^2=[_1,_0,_0,_0]
[1,_1,_0,0]___[_2,_1,_0,_0]
[1,_3,_1,0]___[_5,_6,_1,_0]
[1,12,_9,1]___[23,51,18,_1]

Links

Crossrefs

Cf. A078121, A078122, A078124, A078125.

Programs

  • Maple
    S:= proc(i, j) option remember;
       add(M(i, k)*M(k, j), k=0..i)
    end:
M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
       add(S(i-1, k)*M(k, j-1), k=0..i-1))
    end:
seq(seq(S(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 27 2015
  • Mathematica
    S[i_, j_] := S[i, j] = Sum[M[i, k]*M[k, j], {k, 0, i}]; M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[S[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[S[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

Formula

M(1, j) = A078125(j), M(j+1, j)=2*3^j.

A125804 Main diagonal of table A125800.

Original entry on oeis.org

1, 2, 12, 238, 15200, 3013980, 1828979530, 3373190565626, 18837339867421686, 317817051628161116674, 16176220447967300610844988, 2481251352301850541661479580329, 1146112129196402690505198891390847384
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Table A125800 is related to partitions into powers of 3; column k of A125800 equals row sums of matrix power A078122^k, where triangle A078122 shifts left one column under matrix cube.

Crossrefs

Cf. A125800, A078122; columns: A078125, A078124, A125801, A125802, A125803; A125805 (antidiagonal sums).

Programs

  • PARI
    a(n)=local(q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^n)[n+1,c+1]))

A125805 Antidiagonal sums of table A125800.

Original entry on oeis.org

1, 2, 4, 10, 41, 361, 7741, 417212, 57581062, 20688363559, 19625079296963, 49742424992663959, 340292157995636104240, 6337196928437059669994069, 323627960380394115802942263514, 45610724032832026072070666274435391
Offset: 0

Views

Author

Paul D. Hanna, Dec 10 2006

Keywords

Comments

Table A125800 is related to partitions into powers of 3; column k of A125800 equals row sums of matrix power A078122^k, where triangle A078122 shifts left one column under matrix cube.

Crossrefs

Cf. A125800, A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal).

Programs

  • PARI
    a(n)=local(q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^(c+1))[n-c+1,1]))

A111842 Row sums of triangle A111840, which shifts columns left and up under matrix cube.

Original entry on oeis.org

1, 2, 7, 46, 595, 16444, 1048303, 162728110, 63746277967, 64594795730680, 172419318632651104, 1229463017642626881490, 23684690483668583872503679, 1244115601652916934000237966330, 179585081405174505374545193721101377
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2005

Keywords

Crossrefs

Cf. A111840, A078125 (variant).

Programs

  • PARI
    {a(n,q=3)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,j]=(A^q)[i-1,1], B[i,j]=(A^q)[i-1,j-1]));));A=B); return(sum(k=0,n,A[n+1,k+1])))}

A346564 Number of compositions (ordered partitions) of 3^n into powers of 3.

Original entry on oeis.org

1, 2, 20, 26426, 61390791862967, 769671787836269530451291677988751813890576
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 23 2021

Keywords

Comments

The next term is too large to include.

Links

Crossrefs

Cf. A000244, A078125, A078932, A248377, A337990, A346565.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[x^(3^k), {k, 0, n}]), {x, 0, 3^n}], {n, 0, 5}]

Formula

a(n) = [x^(3^n)] 1 / (1 - Sum_{k>=0} x^(3^k)).
a(n) = A078932(A000244(n)).
Previous Showing 11-19 of 19 results.

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