A002577 -id:A002577 - cp's OEIS frontend

A111827 Number of partitions of 6^n into powers of 6, also equals the row sums of triangle A111825, which shifts columns left and up under matrix 6th power.

1, 2, 8, 134, 10340, 3649346, 6188114528, 52398157106366, 2277627698797283420, 518758596372421679994170, 628925760908337480420110203736, 4109478867142143642923124190955500214

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..52

Cf. A111825, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111832 (q=7), A111837 (q=8). Column 6 of A145515.

a(n,q=6)=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])))

a(n) = [x^(6^n)] 1/Product_{j>=0}(1-x^(6^j)).

A111832 Number of partitions of 7^n into powers of 7, also equals the row sums of triangle A111830, which shifts columns left and up under matrix 7th power.

Original entry on oeis.org

1, 2, 9, 205, 24901, 16077987, 58169810617, 1226373476385199, 154912862345527456431, 119679779055077323244243580, 574461679441277269788798742908435, 17346328772332966415272910459727649244337, 3328366331331467859745524303574824288197338547909

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..28

Cf. A111830, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111827 (q=6), A111837 (q=8). Column 7 of A145515.

a(n,q=7)=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])))

a(n) = [x^(7^n)] 1/Product_{j>=0}(1-x^(7^j)).

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

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..45

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

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])))

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

Alois P. Heinz, Oct 11 2008

nonn

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

			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].

Alois P. Heinz, Table of n, a(n) for n = 0..45

Cf. 10th column of A145515, A007318.

Cf. A011557, A002577, A078125.

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

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);

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 *)

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

A002576 Coefficients of Bell's formula for making change.

Original entry on oeis.org

2, 16, 130, 1424, 23682, 637328, 28867714, 2260015504, 311718542466, 76844461332880, 34239915581996162, 27825107366882974096, 41547917209230771715202, 114704977949192346233608592, 588650824552337332645472468098

Offset: 3

N. J. A. Sloane

nonn

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [ Swedish ], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103.
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).

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy]

Cf. A002575, A002577.

A diagonal of A262554.

More terms from Sean A. Irvine, Oct 19 2015

A172288 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^2^n into powers of 2 less than or equal to 2^k.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 4, 9, 1, 2, 4, 25, 129, 1, 2, 4, 35, 4225, 32769, 1, 2, 4, 36, 47905, 268468225, 2147483649, 1, 2, 4, 36, 222241, 733276217345, 1152921506754330625, 9223372036854775809, 1

Offset: 0

Alois P. Heinz, Jan 26 2011

A(18,18) = 2797884726...4715787265 has 1420371 decimal digits and was computed by the algorithm given below.

			A(2,1) = 9, because there are 9 partitions of 2^2^2=16 into powers of 2 less than or equal to 2^1=2: [2,2,2,2,2,2,2,2], [2,2,2,2,2,2,2,1,1], [2,2,2,2,2,2,1,1,1,1], [2,2,2,2,2,1,1,1,1,1,1], [2,2,2,2,1,1,1,1,1,1,1,1], [2,2,2,1,1,1,1,1,1,1,1,1,1], [2,2,1,1,1,1,1,1,1,1,1,1,1,1], [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1,     2,         2,            2,               2,  ...
  1,     3,         4,            4,               4,  ...
  1,     9,        25,           35,              36,  ...
  1,   129,      4225,        47905,          222241,  ...
  1, 32769, 268468225, 733276217345, 751333186150401,  ...

Alois P. Heinz, Table of n, a(n) for n = 0..77

Cf. A002577, A000123, A181322, A145515.

Main diagonal gives: A182135.

b:= proc(n,j) option remember; local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(2^(2^n-k), k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n < 0, 0, j == 0, 1, j == 1, n+1, n < j , b[n, j] = b[n-1, j] + b[2*n, j-1] , True, nn = 1 + Floor[n]; r := n - nn; (nn-j)*Binomial[nn, j] * Sum [Binomial[j, h] /(nn - j + h) * b[j - h + r, j] *(-1)^h, {h, 0, j-1}] ] ]; a[n_, k_] := b[2^(2^n-k), k]; Table[Table[a[n, d-n] // FullSimplify, {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

A(n,k) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..k-1} (1-x^(2^j)).

A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

Original entry on oeis.org

1, 3, 7, 16, 40, 119, 450, 2253, 15207, 139190, 1731703, 29335875, 677864041, 21400069232, 924419728471, 54716596051100, 4443400439075834, 495676372493566749, 76041424515817042402, 16060385520094706930608, 4674665948889147697184915

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.

			The a(1) = 1 through a(4) = 16 partitions of partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (4)(1111)
                             (11111111)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (2)(2)(2)(11)
                             (2)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Seiichi Manyama, Table of n, a(n) for n = 1..120

Cf. A000123, A001970, A002577, A006171, A007716, A034729, A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.

Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]

a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ Michel Marcus, Jan 28 2019

A323775 a(n) = Sum_{k = 1...n} k^(2^(n - k)).

Original entry on oeis.org

1, 3, 8, 30, 359, 72385, 4338080222, 18448597098193762732, 340282370354622283774333836315916425069, 115792089237316207213755562747271079374483128445080168204415615259394085515423

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of ways to choose a constant integer partition of each part of a constant integer partition of 2^(n - 1).

			The a(1) = 1 through a(4) = 30 twice-partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (11)(2)       (22)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (1111)(4)
                             (4)(1111)
                             (11111111)
                             (1111)(22)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (11)(2)(2)(2)
                             (2)(11)(2)(2)
                             (2)(2)(11)(2)
                             (2)(2)(2)(11)
                             (11)(11)(2)(2)
                             (11)(2)(11)(2)
                             (11)(2)(2)(11)
                             (2)(11)(11)(2)
                             (2)(11)(2)(11)
                             (2)(2)(11)(11)
                             (11)(11)(11)(2)
                             (11)(11)(2)(11)
                             (11)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Cf. A000123, A001970, A002577, A006171, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323776.

Mathematica
```
Table[Sum[k^2^(n-k),{k,n}],{n,12}]
```

A125799 Antidiagonal sums of table A125790.

Original entry on oeis.org

1, 2, 4, 9, 25, 94, 520, 4521, 64793, 1581010, 67106004, 5029631745, 673439168257, 162631617757086, 71416302988324776, 57430160224301687377, 85096038984339418975505, 233592305902515392375925762, 1193627868786115606927913952196, 11402285904243733254203516140245465

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795, A125796; diagonals: A125797, A125798.

a(n)=local(q=2,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]))

A212775 Number of partitions of 2^(2^n) into powers of 2.

Original entry on oeis.org

2, 4, 36, 692004, 114788185359199234852802340

Offset: 0

Alois P. Heinz, May 26 2012

nonn

Lengths (in decimal digits) of the terms a(0), a(1), ... are: 1, 1, 2, 6, 27, 119, 525, 2241, 9330, ... .

			a(0) = 2 because the number of partitions of 2^2^0 = 2 into powers of 2 is 2: [2], [1,1].
a(1) = 4: [4], [2,2], [2,1,1], [1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..6

Cf. A000123, A002577, A172288, A182135.

b:= proc(n, j) option remember; local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(1, 2^n):
seq(a(n), n=0..6);

b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n<0, 0, j==0, 1, j==1, n+1, n < j, b[n, j] = b[n-1, j] + b[2*n, j-1], True, nn = 1+Floor[n]; r = n-nn; (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)*b[j-h+r, j]*(-1)^h, {h, 0, j-1}]]]; a[n_] := b[1, 2^n]; Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Feb 28 2017, translated from Maple *)

a(n) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..2^n-1} (1-x^(2^j)).

cp's OEIS Frontend

A111827 Number of partitions of 6^n into powers of 6, also equals the row sums of triangle A111825, which shifts columns left and up under matrix 6th power.

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A111832 Number of partitions of 7^n into powers of 7, also equals the row sums of triangle A111830, which shifts columns left and up under matrix 7th power.

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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.

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

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A002576 Coefficients of Bell's formula for making change.

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A172288 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^2^n into powers of 2 less than or equal to 2^k.

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A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

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A323775 a(n) = Sum_{k = 1...n} k^(2^(n - k)).

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