A117506 -id:A117506 - cp's OEIS frontend

A134145 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.

1, 3, 1, 15, 3, 1, 105, 15, 9, 3, 1, 945, 105, 45, 15, 9, 3, 1, 10395, 945, 315, 225, 105, 45, 27, 15, 9, 3, 1, 135135, 10395, 2835, 1575, 945, 315, 225, 135, 105, 45, 27, 15, 9, 3, 1, 2027025, 135135, 31185, 14175, 11025, 10395, 2835, 1575, 945, 675, 945, 315, 225

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(3) = A134144 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(3)/M_3.

			[1]; [3,1]; [15,3,1]; [105,15,9,3,1]; [945,105,45,15,9,3,1]; ...
a(4,3)=9 from the third (k=3) partition (2^2) of 4: (3)^2 = 9, because S2(3,2,1) = 3!! = 1*3 = 3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134147 (row sums, also of triangle A134146).

a(n,k) = Product_{j=1..n} S2(3,j,1)^e(n,k,j) with S2(3,n,1) = A035342(n,1) = A001147(n) = (2*n-1)!! and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134144(n,k)/A036040(n,k) (division of partition arrays M_3(3) by M_3).

A134149 A certain partition array in Abramowitz-Stegun (A-St) order.

Original entry on oeis.org

1, 4, 1, 28, 12, 1, 280, 112, 48, 24, 1, 3640, 1400, 1120, 280, 240, 40, 1, 58240, 21840, 16800, 7840, 4200, 6720, 960, 560, 720, 60, 1, 1106560, 407680, 305760, 274400, 76440, 117600, 54880, 47040, 9800, 23520, 6720, 980, 1680, 84, 1, 24344320

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(4), the k=4 member of a family of generalizations of the multinomial number array M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(4,n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3= A036040.
a(n,k) enumerates unordered forests of increasing quaternary trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition.

			[1]; [4,1]; [28,12,1]; [280,112,48,24,1]; [3640,1400,1120,280,240,40,1]; ...
a(4,3)=48 from the third (k=3) partition (2^2) of 4: 4!*((4/2!)^2)/2 = 48, because S2(4,2,1) = 4!!! = 4*1 = 4.
There are a(4,3) = 48 = 3*4^2 unordered 2-forests with 4 vertices, composed of two increasing quaternary (4-ary) trees, each with two vertices: there are 3 increasing labelings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in four versions from the quaternary structure.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134144 (M_3(3) array).

a(n,k) = n!*Product_{j=1..n} (S2(4,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! (triple- or 3-factorials) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

A134150 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(4)/M_3.

Original entry on oeis.org

1, 4, 1, 28, 4, 1, 280, 28, 16, 4, 1, 3640, 280, 112, 28, 16, 4, 1, 58240, 3640, 1120, 784, 280, 112, 64, 28, 16, 4, 1, 1106560, 58240, 14560, 7840, 3640, 1120, 784, 448, 280, 112, 64, 28, 16, 4, 1, 24344320, 1106560, 232960, 101920, 78400, 58240, 14560, 7840

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(4) = A134149 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short, M_3(4)/M_3.

			Triangle begins:
  [1];
  [4,1];
  [28,4,1];
  [280,28,16,4,1];
  [3640,280,112,28,16,4,1];
  ...
a(4,3)=16 from the third (k=3) partition (2^2) of 4: (4)^2 = 16, because S2(4,2,1) = 4!! = 4*1 = 4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

Cf. A134145 (M_3(3)/M_3 array).

Cf. A134152 (row sums, also of triangle A134151).

a(n,k) = Product_{j=1..n} S2(4,j,1)^e(n,k,j) with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134149(n,k)/A036040(n,k) (division of partition arrays M_3(4) by M_3).

A134273 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).

Original entry on oeis.org

1, 5, 1, 45, 15, 1, 585, 180, 75, 30, 1, 9945, 2925, 2250, 450, 375, 50, 1, 208845, 59670, 43875, 20250, 8775, 13500, 1875, 900, 1125, 75, 1, 5221125, 1461915, 1044225, 921375, 208845, 307125, 141750, 118125, 20475, 47250, 13125, 1575, 2625, 105, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(5), the k=5 member in the family of a generalization of the multinomial number arrays M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(5,n,m):=A049029(n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3 = A036040.
a(n,k) enumerates unordered forests of increasing quintic (5-ary) trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition.

			Triangle begins:
  [1];
  [51];
  [45,15,1];
  [585,180,75,30,1];
  [9945,2925,2250,450,375,50,1];
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

Cf. There are a(4, 3)=75=3*5^2 unordered 2-forest with 4 vertices, composed of two 5-ary increasing trees, each with two vertices: there are 3 increasing labelings (1, 2)(3, 4); (1, 3)(2, 4); (1, 4)(2, 3) and each tree comes in five versions from the 5-ary structure.
Cf. A049120 (row sums also of triangle A049029).
Cf. A134149 (M_3(4) array).

a(n,k) = n!*Product_{j=1..n} (S2(5,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-factorials) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

A134274 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5)/M_3.

Original entry on oeis.org

1, 5, 1, 45, 5, 1, 585, 45, 25, 5, 1, 9945, 585, 225, 45, 25, 5, 1, 208845, 9945, 2925, 2025, 585, 225, 125, 45, 25, 5, 1, 5221125, 208845, 49725, 26325, 9945, 2925, 2025, 1125, 585, 225, 125, 45, 25, 5, 1, 151412625, 5221125, 1044225, 447525, 342225

Offset: 1

Wolfdieter Lang, Nov 13 2007

Partition number array M_3(5) = A134273 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(5)/M_3.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

			Triangle begins:
  [1];
  [5,1];
  [45,5,1];
  [585,45,25,5,1];
  [9945,585,225,45,25,5,1];
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

Row sums A134276 (also of triangle A134275).

Cf. A134150 (M_3(4)/M_3 array).

a(n,k) = Product_{j=1..n} S2(5,j,1)^e(n,k,j) with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-factorials) and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134273(n,k)/A036040(n,k) (division of partition arrays M_3(5) by M_3).

A134286 Characteristic sequence for sequence A026905.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Wolfdieter Lang, Nov 13 2007

This partition array is the member k=1 in the family M_0(k), with M_0(2)=M_0= A048996, M_0(3)= A134283, etc.

When read as partition array (tabf with sequence of row lengths given by the partition numbers A000041) in Abramowitz-Stegun order (see A117506 for the reference) a(n,k) is the characteristic partition array for the partition (1^n) of n.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for characteristic functions

terms = 105; nmax = 10;
pp = PartitionsP[Range[nmax]] // Accumulate;
a[n_] := If[n > pp[[-1]], Print["nmax = ", nmax, " too small"], Boole[ MemberQ[ pp, n]]];
Array[a, terms] (* Jean-François Alcover, Jun 19 2019 *)

a(n)=1 if n from A026905, else 0.

A134279 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6)/M_3.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 66, 36, 6, 1, 22176, 1056, 396, 66, 36, 6, 1, 576576, 22176, 6336, 4356, 1056, 396, 216, 66, 36, 6, 1, 17873856, 576576, 133056, 69696, 22176, 6336, 4356, 2376, 1056, 396, 216, 66, 36, 6, 1, 643458816, 17873856, 3459456, 1463616

Offset: 1

Wolfdieter Lang, Nov 13 2007

Partition number array M_3(6) = A134278 with each entry divided by the corresponding one of the partition number array M_3 = M_3(1) = A036040; in short M_3(6)/M_3.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

			[1]; [6,1]; [66,6,1]; [1056,66,36,6,1]; [22176,1056,396,66,36,6,1]; ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Row sums give A134281 (also of triangle A134280).

Cf. A134274 (M_3(5)/M_3 partition array).

a(n,k) = Product_{j=1..n} S2(6,j,1)^e(n,k,j) with S2(6,n,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintuple- or 5-factorials) and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134278(n,k)/A036040(n,k) (division of partition arrays M_3(6) by M_3).

A134283 A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 20, 9, 9, 1, 126, 70, 60, 30, 27, 12, 1, 462, 252, 210, 100, 105, 180, 27, 40, 54, 15, 1, 1716, 924, 756, 700, 378, 630, 300, 270, 140, 360, 108, 50, 90, 18, 1, 6435, 3432, 2772, 2520, 1225, 1386, 2268, 2100, 945, 900, 504, 1260, 600, 1080, 81

Offset: 1

Wolfdieter Lang, Nov 13 2007

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_0(3); the k=3 member in the family of a generalization of the multinomial number arrays M_0 = M_0(2) = A048996.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The s2(3,n,m):=A035324(n,m) numbers (generalized Pascal triangle) are obtained by summing in row n all numbers with the same part number m. In the same manner the s2(2,n,m) = binomial(n-1,m-1) = A007318(n-1,m-1) numbers are obtained from the partition array M_0 = A048996.

			[1]; [3,1]; [10,6,1]; [35,20,9,9,1]; [126,70,60,30,27,12,1]; ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A049027 (row sums, also of triangle A035324).

a(n,k) = m!*Product_{j=1..n} (s2(3,j,1)^e(n,k,j))/e(n,k,j)! with s2(3,n,1) = A035324(n,1) = A001700(n-1) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.

A263003 Partition array for the products of the hook lengths of Ferrers (Young) diagrams corresponding to the partitions of n, written in Abramowitz-Stegun order.

Original entry on oeis.org

1, 1, 2, 2, 6, 3, 6, 24, 8, 12, 8, 24, 120, 30, 24, 20, 24, 30, 120, 720, 144, 80, 144, 72, 45, 144, 72, 80, 144, 720, 5040, 840, 360, 360, 336, 144, 240, 240, 252, 144, 360, 336, 360, 840, 5040, 40320, 5760, 2016, 1440, 2880, 1920, 630, 576, 720, 960, 1152, 448, 720, 576, 2880, 1152, 630, 1440, 1920, 2016, 5760, 40320, 362880, 45360, 13440, 7560, 8640, 12960, 3456, 2240, 4320, 3024, 2160, 8640, 6480, 1920, 1680, 1680, 2160, 4320, 5184, 1920, 3024, 2240, 8640, 6480, 3456, 7560, 12960, 13440, 45360, 362880

Offset: 0

Wolfdieter Lang, Oct 09 2015

The sequence of row lengths is A000041: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...] (partition numbers p(n)).
For the ordering of this tabf array a(n,k) see Abramowitz-Stegun (A-St) ref. pp. 831-2.
This is the array n!/A117506(n,k).
For rows 1..15 of this irregular triangle see the W. Lang link.
The row sums give A263004.
The formula given below is the one obtained from the version given, e.g., in Wybourne's book for A117506(n, k). See also the Glass-Ng reference, Theorem 1, p. 701, which gives the same formula, after rewriting using also a Vandermonde determinant.
In A. Young's third paper (Q.S.A. III, see A117506), Theorem V on p. 266, CP p. 363, f/n! (the present 1/a(n,k)) appears in the decomposition of 1 for each n, that is Sum_{k = 1..p(n)} 1/a(n,k) Sum_{j=1..d(n,k)} Y'(n,k,j) = 1, with d(n,k) = A117506(n,k), and the Young operators Y' for the standard tableaux for the k-th partition of n in A-St order.
a(n,k) also appears as normalization to obtain the idempotents NP/a(n,k). See A. Young, Q.S.A. II, p. 366, CP p. 97: NP = (1/a(n,k)) (NP)^2 for each Young tableau of the shape given by the k-th partition of n in A-St order.

			The first rows of this irregular triangle are:
n\k   1    2    3    4    5   6    7   8   9   10   11
0:    1
1:    1
2:    2    2
3:    6    3    6
4:   24    8   12    8   24
5:  120   30   24   20   24  30  120
6:  720  144   80  144   72  45  144  72  80  144  720
...
Note that the rows are in general not symmetric.
See the W. Lang link for rows n = 1..15.
a(6,6) is related to the (self-conjugate) partition (1, 2, 3) of n = 6, taken in reverse order (3, 2, 1) with the Ferrers (or Young) diagram
   _ _ _
  |_|_|_| and the hook length numbers   5  3  1 ...
  |_|_|                                 3  1
  |_|                                   1
The product gives 5*3*1*3*1*1 = 45 = a(6,6).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
B. Wybourne, Symmetry principles and atomic spectroscopy, Wiley, New York, 1970, p. 9.

Alois P. Heinz, Rows n = 0..30, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Kenneth Glass and Chi-Keung Ng, A Simple Proof of the Hook Length Formula, Am. Math. Monthly 111 (2004) 700 - 704.
Wolfdieter Lang, Rows 1..15.

Cf. A117506, A263004.

h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),
             k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)->`if`(n=0 or i=1, [h([l[], 1$n])],
               `if`(i<1, [], [g(n, i-1, l)[],
               `if`(i>n, [], g(n-i, i, [l[], i]))[]])):
T:= n-> g(n$2, [])[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 05 2015

a(n,k) = Product_{i=1..m(n,k)} (x_i)!/Det(x_i^(m(n,k) - j)) with the Vandermonde determinant for the variables x_i := lambda(n,k)_i + m(n,k) - i, for i, j = 1..m(n,k), where m(n,k) is the number of parts of the k-th partition of n denoted by lambda(n,k), in the A-St order (see above). Lambda(n,k)_i stands for the i-th part of the partition lambda(n,k), sorted in nonincreasing order (this is the reverse of the A-St notation for a partition).

Row n=0 prepended by Alois P. Heinz, Nov 05 2015

A263004 Row sums of the partition array for the products of the hook lengths numbers of Ferrers (or Young) diagrams A263003.

Original entry on oeis.org

1, 1, 4, 15, 76, 368, 2365, 14892, 116236, 966064, 9256889, 96638496, 1129309316, 14261533248, 196315312964, 2900635720869, 45926240752560, 773725147192412, 13831256551416480, 261227089570409028, 5198858467673903360, 108706624576630569271

Offset: 0

Wolfdieter Lang, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A066183, A117506, A263003.

h:= l-> (n-> mul(mul(1+l[i]-j+add(`if`(l[k]>=j, 1, 0),
             k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
                `if`(i>n, 0, g(n-i, i, [l[], i]))+g(n, i-1, l))):
a:= n-> g(n$2, []):
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 05 2015

a(n) = Sum_{k=1..A000041(n)} A263003(n,k).

a(0)=1 prepended by Alois P. Heinz, Nov 05 2015

cp's OEIS Frontend

A134145 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(3)/M_3.

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A134149 A certain partition array in Abramowitz-Stegun (A-St) order.

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A134150 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(4)/M_3.

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A134273 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).

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A134274 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5)/M_3.

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A134286 Characteristic sequence for sequence A026905.

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A134279 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6)/M_3.

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A134283 A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).

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A263003 Partition array for the products of the hook lengths of Ferrers (Young) diagrams corresponding to the partitions of n, written in Abramowitz-Stegun order.

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