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A259095 Triangle read by rows: T(n,r) = number of arrangements of n pennies in rows, with r contiguous pennies in the bottom row, and each higher row consisting of contiguous pennies, each touching two pennies in the row below (1 <= r <= n).

%I A259095 #46 Mar 01 2025 22:49:34
%S A259095 1,0,1,0,1,1,0,0,2,1,0,0,1,3,1,0,0,1,2,4,1,0,0,0,3,3,5,1,0,0,0,2,5,4,
%T A259095 6,1,0,0,0,1,5,7,5,7,1,0,0,0,1,5,8,9,6,8,1,0,0,0,0,4,10,11,11,7,9,1,0,
%U A259095 0,0,0,3,11,15,14,13,8,10,1,0,0,0,0,2,9,19,20,17,15,9,11,1,0,0,0,0,1,9,20,27,25,20,17,10,12,1,0,0,0,0,1,7,20,32,35,30,23,19,11,13,1
%N A259095 Triangle read by rows: T(n,r) = number of arrangements of n pennies in rows, with r contiguous pennies in the bottom row, and each higher row consisting of contiguous pennies, each touching two pennies in the row below (1 <= r <= n).
%C A259095 Computed by _R. K. Guy_ (see link).
%H A259095 Joerg Arndt, <a href="/A259095/b259095.txt">Table of n, a(n) for n = 1..5050</a> (rows 1..100, flattened)
%H A259095 Joerg Arndt, <a href="http://jjj.de/fxt/demo/seq/#A259095">C++ program for this sequence</a>, 2016
%H A259095 F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100027134">On some new types of partitions associated with generalized Ferrers graphs</a>, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
%H A259095 F. C. Auluck, <a href="/A001524/a001524.pdf">On some new types of partitions associated with generalized Ferrers graphs</a> (annotated scanned copy)
%H A259095 R. K. Guy, <a href="/A259095/a259095.pdf">Letter to N. J. A. Sloane, Apr 08 1988</a> (annotated scanned copy, included with permission)
%F A259095 T(n,r) = Sum_{D(n,r)} Product_{k=2..m} abs(p[k]-p[k-1]) where the sum ranges over all partitions of n into distinct parts with maximal part r and the product over the m-1 pairs of successive parts; m is the number of parts in the partition under consideration. [_Joerg Arndt_, Apr 09 2016]
%e A259095 Triangle begins:
%e A259095   1,
%e A259095   0,1,
%e A259095   0,1,1,
%e A259095   0,0,2,1,
%e A259095   0,0,1,3,1,
%e A259095   0,0,1,2,4,1,
%e A259095   0,0,0,3,3,5,1,
%e A259095   0,0,0,2,5,4,6,1,
%e A259095   0,0,0,1,5,7,5,7,1,
%e A259095   0,0,0,1,5,8,9,6,8,1,
%e A259095   0,0,0,0,4,10,11,11,7,9,1,
%e A259095   0,0,0,0,3,11,15,14,13,8,10,1,
%e A259095   0,0,0,0,2,9,19,20,17,15,9,11,1,
%e A259095   0,0,0,0,1,9,20,27,25,20,17,10,12,1,
%e A259095   0,0,0,0,1,7,20,32,35,30,23,19,11,13,1,
%e A259095   ...
%e A259095 (An unusually large number of rows are shown in order to explain the related sequences A005575-A005578.)
%p A259095 b:= proc(n, i, d) option remember; `if`(i*(i+1)/2<n, 0,
%p A259095      `if`(n=0, 1, b(n, i-1, d+1)+`if`(i>n, 0, d*b(n-i, i-1, 1))))
%p A259095     end:
%p A259095 T:= (n, r)-> b(n-r, r-1, 1):
%p A259095 seq(seq(T(n,r), r=1..n), n=1..15);  # _Alois P. Heinz_, Jul 08 2016
%t A259095 b[n_, i_, d_] := b[n, i, d] = If[i*(i+1)/2 < n, 0, If[n == 0, 1, b[n, i-1, d+1] + If[i > n, 0, d*b[n-i, i-1, 1]]]];
%t A259095 T[n_, r_] := b[n-r, r-1, 1];
%t A259095 Table[T[n, r], {n, 1, 15}, {r, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 27 2016, after _Alois P. Heinz_ *)
%Y A259095 Cf. A001524 (row sums), A001519 (column sums).
%Y A259095 Cf. also A005575 (a diagonal), A005576, A005577 (row maxima), A005578.
%K A259095 nonn,tabl
%O A259095 1,9
%A A259095 _N. J. A. Sloane_, Jun 19 2015