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A242642 Triangle read by rows: T(s,n) (n>=1, 1 <= s <= n) = number of s-line partitions of n. 2
1, 2, 3, 3, 5, 6, 5, 10, 12, 13, 7, 16, 21, 23, 24, 11, 29, 40, 45, 47, 48, 15, 45, 67, 78, 83, 85, 86, 22, 75, 117, 141, 152, 157, 159, 160, 30, 115, 193, 239, 263, 274, 279, 281, 282, 42, 181, 319, 409, 457, 481, 492, 497, 499, 500, 56, 271, 510, 674, 768, 816, 840, 851, 856, 858, 859 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..200, flattened

P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II.

EXAMPLE

Triangle begins:

[1]

[2, 3]

[3, 5, 6]

[5, 10, 12, 13]

[7, 16, 21, 23, 24]

[11, 29, 40, 45, 47, 48]

[15, 45, 67, 78, 83, 85, 86]

[22, 75, 117, 141, 152, 157, 159, 160]

...

The square array (A242641 with n=0 column omitted) begins:

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...

1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, ...

1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, ...

1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, ...

1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, ...

1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, ...

1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, ...

...

MAPLE

T:= proc(s, n) option remember; `if`(n=0, 1, add(add(min(d, s)

      *d, d=numtheory[divisors](j))*T(s, n-j), j=1..n)/n)

    end:

seq(seq(T(s, n), s=1..n), n=1..14);  # Alois P. Heinz, Oct 02 2018

MATHEMATICA

T[s_, n_] := T[s, n] = If[n==0, 1, Sum[Sum[Min[d, s]*d, {d, Divisors[j]}]* T[s, n - j], {j, 1, n}]/n];

Table[Table[T[s, n], {s, 1, n}], {n, 1, 14}] // Flatten (* Jean-Fran├žois Alcover, Dec 10 2019, after Alois P. Heinz *)

CROSSREFS

Upper triangle of array in A242641 (with the n=0 column omitted).

Sequence in context: A183322 A295918 A296834 * A178041 A181805 A212010

Adjacent sequences:  A242639 A242640 A242641 * A242643 A242644 A242645

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, May 21 2014

STATUS

approved

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Last modified March 8 00:38 EST 2021. Contains 341934 sequences. (Running on oeis4.)