A096651 -id:A096651 - cp's OEIS frontend

A096800 Triangle of coefficients, read by row polynomials P_n(y), that satisfy the g.f.: A096651(x,y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], with P_n(0)=0 for n>=1.

1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 4, -5, 5, 0, 1, 2, 2, -5, 6, 0, 1, 6, -28, 28, -7, 7, 0, 1, 4, 90, -136, 49, -8, 8, 0, 1, 6, -738, 1082, -432, 90, -9, 9, 0, 1, 4, 6279, -9525, 4075, -969, 145, -10, 10, 0, 1, 10, -66594, 101915, -44803, 11143, -1881, 220, -11, 11, 0, 1, 4, 816362, -1260268, 565988, -144300, 25207, -3300, 318

Offset: 0

Paul D. Hanna, Jul 13 2004

Row sums form the positive integers. The first column forms the totients (A000010). The inverse Moebius transform of each column forms the columns of triangle {n/k*A096799(n,k)}. A generalized Euler transform of the row polynomials of this triangle generates A096651; the row sums of A096651^n form the n-dimensional partitions.

			G.f.: 1/A096651(x,y) = (1-x)^y*(1-x^2)^[(y+y^2)/2]*(1-x^3)^[(2y+y^3)/3]*(1-x^4)^[(2y+y^2+y^4)/4]*(1-x^5)^[(4y-5y^2+5y^3+y^5)/5]*...
Rows begin:
[1],
[1,1],
[2,0,1],
[2,1,0,1],
[4,-5,5,0,1],
[2,2,-5,6,0,1],
[6,-28,28,-7,7,0,1],
[4,90,-136,49,-8,8,0,1],
[6,-738,1082,-432,90,-9,9,0,1],
[4,6279,-9525,4075,-969,145,-10,10,0,1],
[10,-66594,101915,-44803,11143,-1881,220,-11,11,0,1],
[4,816362,-1260268,565988,-144300,25207,-3300,318,-12,12,0,1],
[12,-11418459,17738565,-8095100,2105129,-375609,50414,-5382,442,-13,13,0,1],...

Cf. A096651, A096799.

A096874 Matrix inverse of triangle A096651; transforms n-dimensional partitions into (n-1)-dimensional partitions.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 1, -1, -1, 1, 0, -1, 3, -2, -1, 1, 0, -1, -1, 5, -3, -1, 1, 0, 3, -5, -1, 7, -4, -1, 1, 0, 4, -10, 4, -2, 9, -5, -1, 1, 0, -17, 121, -146, 42, -5, 11, -6, -1, 1, 0, -27, -662, 1155, -591, 130, -11, 13, -7, -1, 1, 0, 118, 5952, -10015, 5327, -1662, 294, -21, 15, -8, -1, 1, 0, 267, -70266, 113346, -57476

Offset: 0

Paul D. Hanna, Jul 13 2004

Row sums form {1,1,0,0,0,...(zeros continue)}.

			Rows begin:
[1],
[0,1],
[0,-1,1],
[0,0,-1,1],
[0,1,-1,-1,1],
[0,-1,3,-2,-1,1],
[0,-1,-1,5,-3,-1,1],
[0,3,-5,-1,7,-4,-1,1],
[0,4,-10,4,-2,9,-5,-1,1],
[0,-17,121,-146,42,-5,11,-6,-1,1],
[0,-27,-662,1155,-591,130,-11,13,-7,-1,1],
[0,118,5952,-10015,5327,-1662,294,-21,15,-8,-1,1],
[0,267,-70266,113346,-57476,17435,-3843,565,-36,17,-9,-1,1],
[0,-917,908722,-1473694,746476,-220017,46320,-7821,979,-57,19,-10,-1,1],...

Cf. A096651.

A096642 Column with index 2 of triangle A096651: a(n) = A096651(n+2,2).

Original entry on oeis.org

1, 1, 2, 1, 3, -1, 15, -78, 632, -6049, 68036, -878337, 12817659, -208803098, 3758607935, -74132077726, 1590647874073

Offset: 0

Paul D. Hanna, Jul 06 2004

sign

The row sums of A096651^n form the n-dimensional partitions.

Cf. A096651.

a(12)-a(16) taken from a096651.txt. M. F. Hasler, Apr 12 2012

A096645 Column with index 5 of triangle A096651: a(n) = A096651(n+5,5).

Original entry on oeis.org

1, 1, 5, 1, 25, -152, 1506, -16668, 214528, -3123249

Offset: 0

Paul D. Hanna, Jul 06 2004

sign

The row sums of A096651^n form the n-dimensional partitions.

Cf. A096651.

A096742 Numerator of a(n)/2^A005187(n-1), the n-th row sums of A096651^(1/2), with a(0)=1.

Original entry on oeis.org

1, 1, 3, 15, 41, 387, 1017, 4715, 11917, 220323, 517545, 2403313, 6436023, 58028007, 53008869

Offset: 0

Paul D. Hanna, Jul 06 2004

The denominators are 2^A005187(n-1) (for n>0), where A005187(n) is the number of 1's in binary expansion of 2n. Can the row sums of A096651^(1/2) be said to define the (1/2)-dimensional partitions of n?

			Sequence begins: {1,1,3/2,15/8,41/16,387/128,1017/256,...}.
Formed from the row sums of triangular matrix A096651^(1/2), which begins:
{1},
{0,1},
{0,1/2,1},
{0,3/8,1/2,1},
{0,3/16,7/8,1/2,1},
{0,27/128,-1/16,11/8,1/2,1},
{0,35/256,99/128,-5/16,15/8,1/2,1},
{0,103/1024,-229/256,267/128,-9/16,19/8,1/2,1},
{0,-129/2048,7011/1024,-2349/256,595/128,-13/16,23/8,1/2,1},...
The denominator of each element at column n, row k, is A005187(n-k).

Cf. A096651, A096743, A005187.

A096743 Numerator of a(n)/2^A005187(n-1), the n-th row sums of A096651^(3/2), with a(0)=1.

Original entry on oeis.org

1, 1, 5, 35, 135, 1755, 6303, 39815, 132675, 3322515, 10561455, 64566253, 199681945, 2391238415, 7233344915

Offset: 0

Paul D. Hanna, Jul 06 2004

The denominators are 2^A005187(n-1) (for n>0), where A005187(n) is the number of 1's in binary expansion of 2n. Can the row sums of A096651^(3/2) be said to define the (3/2)-dimensional partitions of n?

			Sequence begins: {1,1,5/2,35/8,135/16,1755/128,6303/256,...}.
Formed from the row sums of triangular matrix A096651^(3/2), which begins:
{1},
{0,1},
{0,3/2,1},
{0,15/8,3/2,1},
{0,41/16,27/8,3/2,1},
{0,387/128,53/16,39/8,3/2,1},
{0,1017/256,987/128,65/16,51/8,3/2,1},
{0,4715/1024,753/256,2067/128,77/16,63/8,3/2,1},
{0,11917/2048,29983/1024,-4503/256,3819/128,89/16,75/8,3/2,1},...
The denominator of each element at column n, row k, is A005187(n-k).

Cf. A096651, A096742, A005187.

a(n)/2^A005187(n-1) = Sum_{k=0..n} A096651(n, k)*A096742(k)/2^A005187(k-1).

A096875 Matrix square of inverse triangle A096651; transforms n-dimensional partitions into (n-2)-dimensional partitions.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 1, -2, 1, 0, 3, -1, -2, 1, 0, -6, 9, -3, -2, 1, 0, -3, -7, 15, -5, -2, 1, 0, 23, -27, -6, 21, -7, -2, 1, 0, 9, 15, -32, -5, 27, -9, -2, 1, 0, -141, 360, -267, 16, -6, 33, -11, -2, 1, 0, -74, -1603, 2691, -1216, 161, -11, 39, -13, -2, 1, 0, 1139, 10961, -20469, 11512, -3489, 457, -22, 45, -15, -2, 1, 0, 1119

Offset: 0

Paul D. Hanna, Jul 13 2004

Also matrix square of triangle A096874. Row sums form A096876.

			Rows begin:
[1],
[0,1],
[0,-2,1],
[0,1,-2,1],
[0,3,-1,-2,1],
[0,-6,9,-3,-2,1],
[0,-3,-7,15,-5,-2,1],
[0,23,-27,-6,21,-7,-2,1],
[0,9,15,-32,-5,27,-9,-2,1],
[0,-141,360,-267,16,-6,33,-11,-2,1],
[0,-74,-1603,2691,-1216,161,-11,39,-13,-2,1],
[0,1139,10961,-20469,11512,-3489,457,-22,45,-15,-2,1],
[0,1119,-140656,226512,-116125,36536,-8079,968,-41,51,-17,-2,1],
[0,-10921,1858877,-2993422,1507887,-444319,95633,-16387,1768,-70,57,-19,-2,1],...

Cf. A096651, A096874, A096876.

A096643 Column with index 3 of triangle A096651: a(n) = A096651(n+3,3).

Original entry on oeis.org

1, 1, 3, 1, 7, -17, 133, -1020, 9826, -110514, 1427046, -20827070

Offset: 0

Paul D. Hanna, Jul 06 2004

sign

The row sums of A096651^n form the n-dimensional partitions.

Cf. A096651.

A096644 Column with index 4 of triangle A096651: a(n) = A096651(n+4,4).

Original entry on oeis.org

1, 1, 4, 1, 14, -61, 529, -4989, 56161, -724881, 10576885

Offset: 0

Paul D. Hanna, Jul 06 2004

sign

The row sums of A096651^n form the n-dimensional partitions.

Cf. A096651.

A007042 Left diagonal of partition triangle A047812.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 20, 28, 40, 54, 75, 99, 133, 174, 229, 295, 383, 488, 625, 790, 1000, 1253, 1573, 1956, 2434, 3008, 3716, 4563, 5602, 6840, 8347, 10141, 12308, 14881, 17975, 21635, 26013, 31183, 37336, 44581, 53172, 63259, 75173, 89132, 105556, 124752

Offset: 1

N. J. A. Sloane, R. K. Guy

For n > 2, a(n) is also the number of partitions of n into parts <= n-2: a(n) = A026820(n+1, n-1). - Reinhard Zumkeller, Jan 21 2010
Also, the number of partitions of 2*n in which n-1 is the maximal part; see the Mathematica section. - Clark Kimberling, Mar 13 2012
This is column 2 of the matrix A in Sect. 2.3 of the Govindarajan preprint, cf. references and A096651. - M. F. Hasler, Apr 12 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Govindarajan, Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A000041, A007044, A007045, A026820, A047812, A051643, A096651.

Column k = 2 of A081719.

using Nemo
function A007042List(len)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(-1, len+2, z)
    [coeff(e, j) - 2 for j in 2:len+1] end
A007042List(45) |> println # Peter Luschny, May 30 2020

f[n_]:= Length[Select[IntegerPartitions[2 n], First[#]==n-1 &]]; Table[f[n], {n, 1, 24}] (* Clark Kimberling, Mar 13 2012 *)
a[n_]:= PartitionsP[n+1]-2; Table[a[n], {n,1,50}] (* Jean-François Alcover, Jan 28 2015, after M. F. Hasler *)

A007042(n)=numbpart(n+1)-2  \\ M. F. Hasler, Apr 12 2012

a(n) = A000041(n+1) - 2. - Vladeta Jovovic, Oct 06 2001

More terms from James Sellers

Name edited by Petros Hadjicostas, May 31 2020

cp's OEIS Frontend

A096800 Triangle of coefficients, read by row polynomials P_n(y), that satisfy the g.f.: A096651(x,y) = Product_{n>=1} 1/(1-x^n)^[P_n(y)/n], with P_n(0)=0 for n>=1.

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A096874 Matrix inverse of triangle A096651; transforms n-dimensional partitions into (n-1)-dimensional partitions.

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A096642 Column with index 2 of triangle A096651: a(n) = A096651(n+2,2).

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A096645 Column with index 5 of triangle A096651: a(n) = A096651(n+5,5).

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A096742 Numerator of a(n)/2^A005187(n-1), the n-th row sums of A096651^(1/2), with a(0)=1.

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A096743 Numerator of a(n)/2^A005187(n-1), the n-th row sums of A096651^(3/2), with a(0)=1.

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A096875 Matrix square of inverse triangle A096651; transforms n-dimensional partitions into (n-2)-dimensional partitions.

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A096643 Column with index 3 of triangle A096651: a(n) = A096651(n+3,3).

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A096644 Column with index 4 of triangle A096651: a(n) = A096651(n+4,4).

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A007042 Left diagonal of partition triangle A047812.

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