A145357 -id:A145357 - cp's OEIS frontend

A145359 Second column (m=2) of triangle A145357 (S1hat(6)).

1, 6, 78, 588, 6804, 62496, 753984, 8273664, 109118016, 1408700160, 20697223680, 309818234880, 5100265336320, 87215844925440, 1605019401216000, 30898776106598400, 631358784683827200, 13503510009903513600, 303781522335363072000, 7137966147222822912000

Offset: 1

Wolfdieter Lang, Oct 17 2008

Cf. A145357, A001725 (first column), A145360 (third column).

a(n) = A145357(n+2,2), n>=0.

Changed offset from 0 to 1 by Wolfdieter Lang, Nov 17 2008

A145358 Row sums of triangle A145357 and of array A145356.

Original entry on oeis.org

1, 7, 49, 421, 3697, 37933, 404341, 4841341, 61291861, 848268421, 12465197269, 197121724549, 3302727019621, 58876330343173, 1107627703783909, 21990846791015845, 458649487144877029, 10035575758396240549, 229687317927141716005, 5489966024995169233573

Offset: 1

Wolfdieter Lang, Oct 17 2008

Cf. A145356, A145357.

a(n) = Sum_{m=1..n} A145357(n,m).

Changed offset from 0 to 1 by Wolfdieter Lang, Nov 17 2008

A145360 Third column (m=3) of triangle A145357 (S1hat(6)).

Original entry on oeis.org

1, 6, 78, 804, 8316, 85176, 1021608, 11394432, 144885888, 1886008320, 26529503616, 390680672256, 6237490904064, 104348756736000, 1873079661358080, 35428981499412480, 710916534432276480, 14999585506406645760, 333292875250857984000, 7754472236069830656000

Offset: 0

Wolfdieter Lang, Oct 17 2008

Cf. A001725 (first column), A145359 (second column).

a(n) = A145357(n+3,3), n>=0.

A145356 Partition number array, called M31hat(6).

Original entry on oeis.org

1, 6, 1, 42, 6, 1, 336, 42, 36, 6, 1, 3024, 336, 252, 42, 36, 6, 1, 30240, 3024, 2016, 1764, 336, 252, 216, 42, 36, 6, 1, 332640, 30240, 18144, 14112, 3024, 2016, 1764, 1512, 336, 252, 216, 42, 36, 6, 1, 3991680, 332640, 181440, 127008, 112896, 30240, 18144, 14112

Offset: 1

Wolfdieter Lang, Oct 17 2008, Oct 28 2008

Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(6;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
Sixth member (K=6) in the family M31hat(K) of partition number arrays.
If M31hat(6;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(6):= A145357.

			Triangle begins
  [1];
  [6,1];
  [42,6,1];
  [336,42,36,6,1];
  [3024,336,252,42,36,6,1];
  ...
a(4,3)= 36 = |S1(6;2,1)|^2. The relevant partition of 4 is (2^2).

Wolfdieter Lang, First 10 rows of the array and more.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Cf. A145358 (row sums).

Cf. A144890 (M31hat(5) array), A145357 (S1hat(6)).

a(n,k) = product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)| = A049374(n,1) = A001725(n+4) = [1,6,42,336,3024,30240,332640,...] = (n+4)!/5!, 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.

cp's OEIS Frontend

A145359 Second column (m=2) of triangle A145357 (S1hat(6)).

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A145358 Row sums of triangle A145357 and of array A145356.

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A145360 Third column (m=3) of triangle A145357 (S1hat(6)).

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A145356 Partition number array, called M31hat(6).

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