A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

1, 1, 4, 5, 11, 16, 36, 65, 142, 285, 595, 1210, 2497, 5134, 10726, 22637, 48383, 104066, 224296, 481985, 1030299, 2188912, 4626313, 9743750, 20492711, 43114180, 90843475, 191776658, 405528200, 858384333, 1817311451, 3845500427, 8129033837, 17162815092

Offset

0,3

Links

Table of n, a(n) for n=0..33.

Example

Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.

Mathematica

nn=30;
q=Table[PartitionsP[n], {n, 0, nn}];
t=Table[Sum[(-1)^(j-k)*Binomial[j, k]*q[[i+k]], {k, 0, j}], {j, 0, Length[q]/2}, {i, Length[q]/2}]
Total/@Abs/@Table[t[[j, i-j+1]], {i, nn/2}, {j, i}]

Crossrefs

These are the antidiagonal-sums of the absolute value of A175804.

First column of the same array is A281425.

For primes we have A376681 or A376684, signed A140119 or A376683.

For composites we have A377035, signed A377034.

For squarefree numbers we have A377040, signed A377039.

For nonsquarefree numbers we have A377048, signed A377049.

For prime powers we have A377053, signed A377052.

The signed version is A377056.

The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.

A000009 counts strict integer partitions, differences A087897, A378972.

A000041 counts integer partitions, differences A002865, A053445.

Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.

Sequence in context: A066898 A118143 A001350 * A077238 A185507 A000286

Adjacent sequences: A378618 A378619 A378620 * A378622 A378625 A378626

Keyword

nonn,new

Author

Gus Wiseman, Dec 14 2024

Status

approved