A134150 -id:A134150 - cp's OEIS frontend

A134151 Triangle of numbers obtained from the partition array A134150.

1, 4, 1, 28, 4, 1, 280, 44, 4, 1, 3640, 392, 44, 4, 1, 58240, 5544, 456, 44, 4, 1, 1106560, 80640, 5992, 456, 44, 4, 1, 24344320, 1519840, 88256, 6248, 456, 44, 4, 1, 608608000, 31420480, 1631392, 90048, 6248, 456, 44, 4, 1, 17041024000, 766525760, 33293120

Offset: 1

Wolfdieter Lang Nov 13 2007

This triangle is named S2(4)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1]; [4,1]; [28,4,1]; [280,44,4,1]; [3640,392,44,4,1];...

W. Lang, First 10 rows and more.

Cf. A134152 (row sums). A134272 (alternating row sums).

Cf. A134146 (S2(3)' triangle).

a(n,m)=sum(product(S2(4;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S2(4;j,1)= A007559(j) = A035469(j,1) = (3*j-2)!!!.

A134152 Row sums of triangle A134151 and partition array A134150 (M3(4)/M3).

Original entry on oeis.org

1, 5, 33, 329, 4081, 64289, 1193697, 25959169, 641756673, 17842602561, 549786056897, 18611202710465, 686107435427009, 27365450091496641, 1173865020528758465, 53888878469193228993, 2635977957955398642369

Offset: 1

Wolfdieter Lang, Nov 13 2007

Cf. A000041, A134272 (alternating row sums of triangle A134151).

a(n) = Sum_{m=1..n} A134151(n,m), n>=1.

a(n) = Sum_{k=1..p(n)} A134150(n,k), with p(n) = A000041(n) (number of partitions of n).

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).

cp's OEIS Frontend

A134151 Triangle of numbers obtained from the partition array A134150.

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A134152 Row sums of triangle A134151 and partition array A134150 (M3(4)/M3).

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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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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula