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

Author

Gus Wiseman, Dec 14 2024

Keywords

• nonn

Examples

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

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.

Programs

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}]