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

%I A378621 #5 Dec 15 2024 23:21:27
%S A378621 1,1,4,5,11,16,36,65,142,285,595,1210,2497,5134,10726,22637,48383,
%T A378621 104066,224296,481985,1030299,2188912,4626313,9743750,20492711,
%U A378621 43114180,90843475,191776658,405528200,858384333,1817311451,3845500427,8129033837,17162815092
%N A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).
%e A378621 Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.
%t A378621 nn=30;
%t A378621 q=Table[PartitionsP[n],{n,0,nn}];
%t A378621 t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}]
%t A378621 Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]
%Y A378621 These are the antidiagonal-sums of the absolute value of A175804.
%Y A378621 First column of the same array is A281425.
%Y A378621 For primes we have A376681 or A376684, signed A140119 or A376683.
%Y A378621 For composites we have A377035, signed A377034.
%Y A378621 For squarefree numbers we have A377040, signed A377039.
%Y A378621 For nonsquarefree numbers we have A377048, signed A377049.
%Y A378621 For prime powers we have A377053, signed A377052.
%Y A378621 The signed version is A377056.
%Y A378621 The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.
%Y A378621 A000009 counts strict integer partitions, differences A087897, A378972.
%Y A378621 A000041 counts integer partitions, differences A002865, A053445.
%Y A378621 Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.
%K A378621 nonn
%O A378621 0,3
%A A378621 _Gus Wiseman_, Dec 14 2024