A134283 -id:A134283 - cp's OEIS frontend

A134286 Characteristic sequence for sequence A026905.

1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Wolfdieter Lang, Nov 13 2007

This partition array is the member k=1 in the family M_0(k), with M_0(2)=M_0= A048996, M_0(3)= A134283, etc.

When read as partition array (tabf with sequence of row lengths given by the partition numbers A000041) in Abramowitz-Stegun order (see A117506 for the reference) a(n,k) is the characteristic partition array for the partition (1^n) of n.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for characteristic functions

terms = 105; nmax = 10;
pp = PartitionsP[Range[nmax]] // Accumulate;
a[n_] := If[n > pp[[-1]], Print["nmax = ", nmax, " too small"], Boole[ MemberQ[ pp, n]]];
Array[a, terms] (* Jean-François Alcover, Jun 19 2019 *)

a(n)=1 if n from A026905, else 0.

A134284 A certain partition array in Abramowitz-Stegun order (A-St order), called M_0(3)/M_0.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 35, 10, 9, 3, 1, 126, 35, 30, 10, 9, 3, 1, 462, 126, 105, 100, 35, 30, 27, 10, 9, 3, 1, 1716, 462, 378, 350, 126, 105, 100, 90, 35, 30, 27, 10, 9, 3, 1, 6435, 1716, 1386, 1260, 1225, 462, 378, 350, 315, 300, 126, 105, 100, 90, 81, 35, 30, 27, 10, 9, 3, 1

Offset: 1

Wolfdieter Lang, Nov 13 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_0(3)= A134283 with each entry divided by the corresponding one of the partition number array M_0 = M_0(2) = A048996; in short M_0(3)/M_0.

			[1]; [3,1]; [10,3,1]; [35,10,9,3,1]; [126,35,30,10,9,3,1]; ...
a(4,3) = 9 = 3^2 because (2^2) is the k=4 partition of n=4 in A-St order and s2(3,2,1)=3.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.

Cf. A134826 (row sums coinciding with those of triangle A134285).

a(n,k) = Product_{j=1..n} s2(3,j,1)^e(n,k,j) with s2(3,n,1) = A035324(n,1) = A001700(n-1) = binomial(2*n-1,n) and with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

a(n,k) = A134283(n,k)/A048996(n,k) (division of partition arrays M_0(3) by M_0).

cp's OEIS Frontend

A134286 Characteristic sequence for sequence A026905.

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