A134275 -id:A134275 - cp's OEIS frontend

A134276 Row sums of triangle A134275 (S2(5)').

1, 6, 51, 661, 10831, 224751, 5523051, 158795026, 5197210126, 191295597726, 7808335521876, 350242714836901, 17115798145640401, 905207663613680151, 51504322711247118276, 3137060525840824654651, 203646443505582301226401

Offset: 1

Wolfdieter Lang, Nov 13 2007

Cf. A000041, A134277 (alternating row sums of triangle A134275).

a(n) = Sum_{m=1..n} A134275(n,m), n>=1.

a(n) = Sum_{k=1..p(n)} A134274(n,k), with p(n) = A000041(n) (number of partitions of n).

A134277 Alternating row sums of triangle A134275 (S2(5)').

Original entry on oeis.org

1, 4, 41, 519, 9201, 194819, 4951381, 144651394, 4811109506, 178860848394, 7364318416206, 332403272058269, 16329673772560231, 867235540509998269, 49517829071816599856, 3024950770151869836519, 196866838411908745248981

Offset: 1

Wolfdieter Lang, Nov 13 2007

Cf. A134276 (row sums of triangle A134275).

a(n) = Sum_{m=1..n} A134275(n,m)*(-1)^(m-1), n>=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).

A134280 Triangle of numbers obtained from the partition array A134279.

Original entry on oeis.org

1, 6, 1, 66, 6, 1, 1056, 102, 6, 1, 22176, 1452, 102, 6, 1, 576576, 32868, 1668, 102, 6, 1, 17873856, 779328, 35244, 1668, 102, 6, 1, 643458816, 23912064, 843480, 36540, 1668, 102, 6, 1, 26381811456, 812173824, 25416072, 857736, 36540, 1668, 102, 6, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(6)'.

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

			[1]; [6,1]; [66,6,1]; [1056,102,6,1]; [22176,1452,102,6,1]; ...

W. Lang, First 10 rows and more.

Cf. A134275 (S2(5)').
Cf. A134281 (row sums).
Cf. A134282 (alternating row sums).

a(n,m)=sum(product(S2(6;j,1)^e(n,m,k,j),j=1..n),k=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,k,j) is the exponent of j in the k-th m part partition of n. S2(6;j,1) = A049385(n,1) = A008548(n) = (5*n-4)(!^5)(quintuple- or 5-factorials).

cp's OEIS Frontend

A134276 Row sums of triangle A134275 (S2(5)').

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A134277 Alternating row sums of triangle A134275 (S2(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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Formula

A134280 Triangle of numbers obtained from the partition array A134279.

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