A088954 -id:A088954 - cp's OEIS frontend

A181322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2*n into powers of 2 less than or equal to 2^k.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 2, 4, 6, 5, 1, 1, 2, 4, 6, 9, 6, 1, 1, 2, 4, 6, 10, 12, 7, 1, 1, 2, 4, 6, 10, 14, 16, 8, 1, 1, 2, 4, 6, 10, 14, 20, 20, 9, 1, 1, 2, 4, 6, 10, 14, 20, 26, 25, 10, 1, 1, 2, 4, 6, 10, 14, 20, 26, 35, 30, 11, 1, 1, 2, 4, 6, 10, 14, 20, 26, 36, 44, 36, 12, 1, 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 56, 42, 13, 1

Offset: 0

Alois P. Heinz, Jan 26 2011

Column sequences converge towards A000123.

			A(3,2) = 6, because there are 6 partitions of 2*3=6 into powers of 2 less than or equal to 2^2=4: [4,2], [4,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1], [1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  ...
  1,  2,  2,  2,  2,  2,  ...
  1,  3,  4,  4,  4,  4,  ...
  1,  4,  6,  6,  6,  6,  ...
  1,  5,  9, 10, 10, 10,  ...
  1,  6, 12, 14, 14, 14,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 4.

Columns k=0-5 give: A000012, A000027(n+1), A002620(n+2), A008804, A088932, A088954.
Main diagonal gives A000123.
Cf. A145515.
See A262553 for another version of this array.
See A072170 for a related array (having the same limiting column).

b:= proc(n, j) local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(n/2^(k-1), k):
seq(seq(A(n, d-n), n=0..d), d=0..13);

b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n<0, 0, j == 0, 1, j == 1, n+1, nJean-François Alcover, Jan 15 2014, translated from Maple *)

A181322(n,k,r=1)={if(nA181322(n-1,k,0)+A181322(2*n,k-1,0),n-=r=1+n\1,(r-k)*binomial(r,k)*sum(i=0,min(k-1,k+n), binomial(k,i)/(r-k+i)*A181322(k-i+n,k,0) *(-1)^i))} \\ From Maple. - M. F. Hasler, Feb 19 2019

G.f. of column k: 1/(1-x) * 1/Product_{j=0..k-1} (1 - x^(2^j)).
A(n,k) = Sum_{i=0..k} A089177(n,i).
For n < 2^k, T(n,k) = A000123(k). T(n,0) = 1, T(n,1) = n+1. - M. F. Hasler, Feb 19 2019

A088932 G.f.: 1/((1-x)^2(1-x^2)(1-x^4)*(1-x^8)).

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, 94, 114, 140, 166, 201, 236, 280, 324, 380, 436, 504, 572, 656, 740, 840, 940, 1060, 1180, 1320, 1460, 1625, 1790, 1980, 2170, 2390, 2610, 2860, 3110, 3396, 3682, 4004, 4326, 4690, 5054, 5460, 5866, 6321, 6776, 7280, 7784

Offset: 0

N. J. A. Sloane, Dec 02 2003

a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^4. First differs from A000123 at n=16. - Alois P. Heinz, Apr 02 2012

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,0,-2,0,2,-2,2,0,-2,1).

See A000027, A002620, A008804, A088954, A000123 for similar sequences.
Column k=4 of A181322.
Cf. A010873.

f := proc(n,k) option remember; if k > n then RETURN(0); fi; if k= 0 then if n=0 then RETURN(1) else RETURN(0); fi; fi; if k = 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(f(n-1,k)); fi; f(n-1,k)+f(n/2,k-1); end; # present sequence is f(2m,5)
GFF := k->x^(2^(k-2))/((1-x)*mul((1-x^(2^j)),j=0..k-2)); # present g.f. is GFF(5)/x^8
a:= proc(n) local m, r; m := iquo(n, 8, 'r'); r:= r+1; [1, 2, 4, 6, 10, 14, 20, 26][r]+ (((8/3*m +(4*r +28)/3)*m +[0, 4, 9, 14, 20, 26, 33, 40][r] +43/3)*m +[22, 33, 50, 67, 93, 119, 154, 189][r]/3)*m end: seq(a(n), n=0..60); # Alois P. Heinz, Apr 17 2009

CoefficientList[Series[1/((1-x)^2(1-x^2)(1-x^4)(1-x^8)), {x,0,60}], x]  (* Harvey P. Dale, Apr 22 2011 *)
Table[1 + 1237*n/1536 + 17*n^2/96 + 13*n^3/768 + n^4/1536 + (5/32 + n/32) * Floor[n/4] + (81/256 + 3*n/32 + n^2/128) * Floor[n/2] - Floor[(n+1)/8]/4 - (n+3) * Floor[(n+1)/4]/32 - Floor[(n+2)/8]/4, {n, 0, 100}] (* Vaclav Kotesovec, May 02 2018 *)
Table[Simplify[1023/1024 + 85*n/96 + 341*n^2/1536 + n^3/48 + n^4/1536 + (-1)^n*(113/1024 + n/32 + n^2/512) - (1 + Sqrt[2])*Cos[Pi*n/4]/16 + Cos[Pi*n/2]/64 + (Sqrt[2] - 1) * Cos[3*Pi*n/4]/16 + (1/8 + n/64)*Sin[Pi*n/2]], {n, 0, 100}] (* Vaclav Kotesovec, May 02 2018 *)

Vec(1/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^8))+O(x^99)) \\ Charles R Greathouse IV, Sep 03 2011

a(n) = (8*floor(n/4)^4 + 8*(m+8)*floor(n/4)^3 - 2*(m^3 - 6*m^2 - 19*m - 86)*floor(n/4)^2 -8*(m^3 - 6*m^2 - 6*m - 22)*floor(n/4) - 7*m^3 + 42*m^2 + 13*m + 54 - (m^3 - 6*m^2 + 5*m + 6)*(-1)^floor(n/4))/48 where m = n mod 4. - Luce ETIENNE, Apr 07 2018

A089177 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n))) giving number of non-squashing partitions of n into k parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 4, 1, 1, 5, 6, 2, 1, 6, 9, 4, 1, 7, 12, 6, 1, 8, 16, 10, 1, 1, 9, 20, 14, 2, 1, 10, 25, 20, 4, 1, 11, 30, 26, 6, 1, 12, 36, 35, 10, 1, 13, 42, 44, 14, 1, 14, 49, 56, 20, 1, 15, 56, 68, 26, 1, 16, 64, 84, 36, 1, 1, 17, 72, 100, 46, 2, 1, 18, 81, 120, 60, 4, 1

Offset: 0

N. J. A. Sloane, Dec 08 2003

T(n,k) = A181322(n,k) - A181322(n,k-1) for n>0. - Alois P. Heinz, Jan 25 2014

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  2;
  1, 4,  4,  1;
  1, 5,  6,  2;
  1, 6,  9,  4;
  1, 7, 12,  6;
  1, 8, 16, 10,  1;

Alois P. Heinz, Rows n = 0..1002, flattened
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.

Cf. A078121, A089178. Columns give A002620, A008804, A088932, A088954. Row sums give A000123.

T:= proc(n) option remember;
     `if`(n=0, 1, zip((x, y)-> x+y, [T(n-1)], [0, T(floor(n/2))], 0)[])
    end:
seq(T(n), n=0..25);  # Alois P. Heinz, Apr 01 2012

row[0] = {1}; row[1] = {1, 1}; row[n_] := row[n] = Plus @@ PadRight[ {row[n-1], Join[{0}, row[Floor[n/2]]]} ]; Table[row[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Jan 31 2014 *)

Row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor(n/2) shifted one place right).
G.f. for column k (k >= 2): x^(2^(k-2))/((1-x)*Product_{j=0..k-2} (1-x^(2^j))). [corrected by Jason Yuen, Jan 12 2025]
Conjecture: let R(n,x) be the n-th reversed row polynomial, then R(n,x) = Sum_{k=0..A000523(A053645(n)) + 1} T(A053645(n),k)*R(2^(A000523(n)-k),x) for n > 0, n != 2^m with R(0,x) = 1 where R(2^m,x) is the (m+1)-th row polynomial of A078121. - Mikhail Kurkov, Jun 28 2025

More terms from Alford Arnold, May 22 2004

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A181322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2*n into powers of 2 less than or equal to 2^k.

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A088932 G.f.: 1/((1-x)^2(1-x^2)(1-x^4)*(1-x^8)).

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A089177 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n))) giving number of non-squashing partitions of n into k parts.

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cp's OEIS Frontend

A181322 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2*n into powers of 2 less than or equal to 2^k.

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Maple

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A088932 G.f.: 1/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^8)).

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Maple

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A089177 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n))) giving number of non-squashing partitions of n into k parts.

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Maple

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Extensions

A088932 G.f.: 1/((1-x)^2(1-x^2)(1-x^4)*(1-x^8)).