A377035 -id:A377035 - cp's OEIS frontend

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

1, 1, 1, 5, 8, 18, 30, 47, 70, 110, 177, 309, 574, 1063, 1892, 3107, 4598, 6166, 8737, 20603, 62457, 149132, 314116, 614093, 1155968, 2176048, 4244322, 8753864, 19006756, 42472117, 95235017, 210396059, 453414950, 949510166, 1931941261, 3826650257, 7400745917

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = 8.

For primes we have A376681 or A376684, signed version A140119 or A376683.
For composites we have A377035, signed version A377034.
For squarefree numbers we have A377040, signed version A377039.
For nonsquarefree numbers we have A377048, signed version A377049.
For prime powers we have A377053, signed version A377052.
For partition numbers we have A378621, signed version A377056.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The signed version is A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244, A325245, A325268.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

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

Original entry on oeis.org

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

Gus Wiseman, Dec 14 2024

nonn

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

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.

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}]

cp's OEIS Frontend

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

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