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%I #64 Apr 22 2019 01:19:42
%S 1,1,2,7,37,268,2496,28612,391189,6230646,113521387,2332049710,
%T 53384167192,1348601249480,37291381915789,1120914133433121,
%U 36406578669907180,1271084987848923282,47487293697623885913,1890771531272515677250,79947079338974990793060
%N Column 1 of triangle A107876.
%C Also number of subpartitions of partition consisting of first n-1 triangular numbers; e.g., a(4) = subp([1,3,6]) = 37. - _Franklin T. Adams-Watters_, Jun 26 2006
%C Number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k) <= s(k-1)+k, see Fxtbook link and example. - _Joerg Arndt_, Apr 30 2011
%C Number of Dyck paths whose ascent lengths are exactly {1,2,...,n+1}; for example, the a(2) = 2 paths are uduuduuudddd and uduudduuuddd. - _David Scambler_, May 30 2012
%C Number of types of cells of a fine mixed subdivision of the Tesler flow polytope. - _Alejandro H. Morales_, Oct 11 2017
%D R. P. Stanley, Enumerative Combinatorics volume 1, 2nd edition, Cambridge University Press, 2011, Ch. 3
%H Alois P. Heinz, <a href="/A107877/b107877.txt">Table of n, a(n) for n = 0..390</a>
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 17.3.6, pp. 368-369
%H K. Mészáros, A. H. Morales, <a href="https://arxiv.org/abs/1710.00701">Volumes and Ehrhart polynomials of flow polytopes</a>, arXiv:1710.00701 [math.CO], 2017, sections 6.1 and 7.
%F G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^(1 + k*(k+1)/2).
%F G.f.: 1 = Sum_{k>=0} a(k)*x^k/(1+x)^((k+1)*(k+2)/2).
%F From _Benedict W. J. Irwin_, Nov 26 2016: (Start)
%F Conjecture: a(n) can be expressed with a series of nested sums,
%F a(3) = Sum_{i=1..2} i+2,
%F a(4) = Sum_{i=1..2} Sum_{j=1..i+2} j+3,
%F a(5) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..j+3} k+4,
%F a(6) = Sum_{i=1..2} Sum_{j=1..i+2} Sum_{k=1..j+3} Sum_{l=1..k+4} l+5. (End)
%F Determinantal formula: a(n) = Det(A) where A is the n X n matrix with entries A(i,j) = binomial(binomial(n+1-i,2)+1,i-j+1). This follows by the formula by MacMahon (see EC1 Ex 3.63) for the number of such subpartitions. - _Alejandro H. Morales_, Aug 31 2017
%e 1 = 1*(1-x)^1 + 1*x*(1-x)^2 + 2*x^2*(1-x)^4 + 7*x^3*(1-x)^7 + 37*x^4*(1-x)^11 + 268*x^5*(1-x)^16 + 2496*x^6*(1-x)^22 + ...
%e Also equals the final term in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated n+1 times, starting with a '1' in row 0, as illustrated by:
%e 1;
%e 1, 1;
%e 1, 2, 2, 2;
%e 1, 3, 5, 7, 7, 7, 7;
%e 1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37;
%e 1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268; ...
%e Restricted growth strings: a(0)=1 corresponds to the empty string; a(1)=1 to [0];
%e a(2) = 2 to [00] and [01]; a(3)=7 to
%e 1: [ 0 0 0 ],
%e 2: [ 0 0 1 ],
%e 3: [ 0 0 2 ],
%e 4: [ 0 1 0 ],
%e 5: [ 0 1 1 ],
%e 6: [ 0 1 2 ],
%e 7: [ 0 1 3 ].
%e [_Joerg Arndt_, Apr 30 2011]
%p b:= proc(n, y) option remember; `if`(n=0, 1, add(
%p b(n-1, y+i-n), i=max(1, n-y)..n*(n-1)/2+1-y))
%p end:
%p a:= n-> b(n+1, 0):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Nov 26 2016
%p # second Maple program:
%p a:= n-> LinearAlgebra:-Determinant(Matrix(n,(i,j)->
%p binomial(binomial(n+1-i,2)+1,i-j+1))):
%p seq(a(n), n=0..25); # _Alejandro H. Morales_, Aug 31 2017
%t a[ n_, k_: 1, j_: 1] := If[ n < 2, Boole[n >= 0], a[n, k, j] = Sum[a[n - 1, i, j + 1], {i, k + j}]]; (* _Michael Somos_, Nov 26 2016 *)
%o (PARI) {a(n)=polcoeff(1-sum(k=0,n-1,a(k)*x^k*(1-x+x*O(x^n))^(1+k*(k+1)/2)),n)}
%Y Cf. A107876, A107878, A107879, A115728, A115729.
%Y Cf. A305605, A305601.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 04 2005, Apr 10 2007
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