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A049611
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a(n) = T(n,2), array T as in A049600.
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20
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0, 1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, 22784, 52736, 120832, 274432, 618496, 1384448, 3080192, 6815744, 15007744, 32899072, 71827456, 156237824, 338690048, 731906048, 1577058304, 3388997632, 7264534528, 15535702016
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OFFSET
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0,3
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COMMENTS
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We may ask for the number of one-step transitions (NOOST) between all unlabeled hierarchies of n elements with the restriction that no subhierarchies are allowed. As an example, consider n = 4 and the hierarchy H1 = [[2,2]] with two elements on level 1 and two on level 2. Starting from H1 the hierarchies [[1, 3]], [[2, 1, 1]], [[1, 2, 1]] can be reached by moving one element only, but [[1, 1, 2]] cannot be reached in a one-step transitition. The solution is n = 1, NOOST = 0; n = 2, NOOST = 1; n = 3, NOOST = 4; n = 4, NOOST = 13; n = 5, NOOST = 38; n = 6, NOOST = 104; n = 7, NOOST = 272; n = 8, NOOST = 688; n = 9, NOOST = 1696. This is sequence A049611. - Thomas Wieder, Feb 28 2004
If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n+1) is the number of (n+2)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
In each composition (ordered partition) of the integer n, circle the first summand once, circle the second summand twice, etc. a(n) is the total number of circles in all compositions of n (that is, add k*(k+1)/2 for each composition into k parts). Note the O.g.f. is B(A(x)) where A(x)= x/(1-x) and B(x)= x/(1-x)^3.
This is the Riordan transform with the Riordan matrix A097805 (of the associated type) of the triangular number sequence A000217. See a Feb 17 2017 comment on A097805. - Wolfdieter Lang, Feb 17 2017
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LINKS
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FORMULA
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G.f.: x*(1-x)^2/(1-2*x)^3.
Binomial transform of quarter squares A002620(n+1): a(n) = Sum_{k=0..n} binomial(n, k)*floor((k+1)^2/4). - Paul Barry, May 27 2003
a(n) = 2^(n-4)*(n^2+5*n+2) - 0^n/8. - Paul Barry, Jun 09 2003
a(n) = Hyper2F1([-n+1, 3], [1], -1) for n>0. - Peter Luschny, Aug 02 2014
a(n) = Sum_{k=0..n-1} Sum_{j=0..n-1} Sum_{i=0..n-1} binomial(n-1, i+j+k). - Yalcin Aktar, Aug 27 2023
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MATHEMATICA
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CoefficientList[Series[x (1-x)^2/(1-2x)^3, {x, 0, 40}], x] (* Harvey P. Dale, Sep 24 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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