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A145362 Lower triangular array S1hat(-1) read by rows, related to partition number array A145361.

%I A145362 #30 May 27 2025 10:20:59
%S A145362 1,1,1,0,1,1,0,1,1,1,0,0,1,1,1,0,0,1,1,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,
%T A145362 1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,
%U A145362 0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,1,1
%N A145362 Lower triangular array S1hat(-1) read by rows, related to partition number array A145361.
%C A145362 If in the partition array M31hat(-1):=A145361 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-1). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C A145362 The first column is [1,1,0,0,0,...]=A008279(1,n-1), n>=1.
%C A145362 T(n,m) gives the number of partitions of n with m parts, with each part not exceeding 2. - _Wolfdieter Lang_, Aug 03 2023
%H A145362 Wolfdieter Lang, <a href="/A145362/a145362.txt">First 10 rows of the array and more</a>.
%H A145362 Louis Comtet, <a href="https://doi.org/10.1007/978-94-010-2196-8">Advanced Combinatorics</a>, Reidel (1974).
%H A145362 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.html">Combinatorial Interpretation of Generalized Stirling Numbers</a>, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
%F A145362 T(n,m) = Sum_{q=1..p(n,m)} Product_{j=1..n} S1(-1;j,1)^e(n,m,q,j) if n>=m>=1, else 0. Here p(n,m) = A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-1;n,1) = A008279(1,n-1) = [1,1,0,0,0,...], n>=1.
%F A145362 The triangle starts in row n with ceiling(n/2) - 1 zeros, and is 1 otherwise. - _Wolfdieter Lang_, Aug 03 2023
%F A145362 G.f.: 1/((1-u*t)*(1-u*t^2)). [Comtet page 97 [2c]]. - _R. J. Mathar_, May 27 2025
%e A145362 Triangle begins:
%e A145362   [1];
%e A145362   [1,1];
%e A145362   [0,1,1];
%e A145362   [0,1,1,1];
%e A145362   [0,0,1,1,1];
%e A145362   [0,0,1,1,1,1];
%e A145362   ...
%o A145362 (PARI) T(n, k) = n<=2*k; \\ _Jinyuan Wang_, Jan 19 2025
%Y A145362 Cf. A004526(n+2), n>=1, (row sums).
%Y A145362 Cf. A008275, A008279, A008284, A036039, A145361, A339884 (parts <=3), A232539 (parts <=4).
%K A145362 nonn,easy,tabl
%O A145362 1,1
%A A145362 _Wolfdieter Lang_, Oct 17 2008