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A007476 Shifts 2 places left under binomial transform.

%I A007476 M1192 #81 Aug 07 2023 08:04:31
%S A007476 1,1,1,2,4,9,23,65,199,654,2296,8569,33825,140581,612933,2795182,
%T A007476 13298464,65852873,338694479,1805812309,9963840219,56807228074,
%U A007476 334192384460,2026044619017,12642938684817,81118550133657,534598577947465,3615474317688778,25070063421597484
%N A007476 Shifts 2 places left under binomial transform.
%C A007476 Starting (1, 2, 4, 9, 23, ...) = row sums of triangle A153859. - _Gary W. Adamson_, Jan 02 2009
%C A007476 Binomial transform of the sequence starting (1, 1, 2, 4, 9, ...) = first differences of (1, 2, 4, 9, 23, ...); that is, (1, 2, 5, 14, 42, 134, 455, 1642, ...). - _Gary W. Adamson_, May 20 2013
%C A007476 Row sums of triangle A256161. - _Margaret A. Readdy_, Mar 16 2015
%C A007476 RG-words corresponding to set partitions of {1, ..., n} with every even entry appearing exactly once. - _Margaret A. Readdy_, Mar 16 2015
%C A007476 a(n) is the number of partitions of [n] whose blocks can be written such that the smallest elements form an increasing sequence and the largest elements form a decreasing sequence. a(5) = 9: 12345, 1235|4, 1245|3, 125|34, 1345|2, 135|24, 145|23, 15|234, 15|24|3. - _Alois P. Heinz_, Apr 24 2023
%D A007476 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007476 Alois P. Heinz, <a href="/A007476/b007476.txt">Table of n, a(n) for n = 0..650</a> (first 101 terms from T. D. Noe)
%H A007476 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
%H A007476 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A007476 Yue Cai and Margaret Readdy, <a href="http://arxiv.org/abs/1506.03249">Negative q-Stirling numbers</a>, arXiv:1506.03249 [math.CO], 2015.
%H A007476 A. Claesson and T. Mansour, <a href="http://arxiv.org/abs/math/0107044">Permutations avoiding a pair of Babson-Steingrimsson patterns</a>, arXiv:math/0107044 [math.CO], 2001-2010.
%H A007476 Rigoberto Flórez, José L. Ramírez, Fabio A. Velandia, and Diego Villamizar, <a href="https://arxiv.org/abs/2308.02059">Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions</a>, arXiv:2308.02059 [math.CO], 2023. See Table 1 p. 13.
%H A007476 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A007476 L. Sze, <a href="https://web.archive.org/web/20050425005413/http://lsze.cosam.calpoly.edu/oeis70.pdf">OEIS conjecture 70</a>
%F A007476 G.f.: Sum_{k>=0} x^(2k)/(Product_{m=0..k-1} (1-mx) * Product_{m=0..k+1} (1-mx)).
%F A007476 G.f. A(x) satisfies A(x) = 1 + x + (x^2/(1-x))*A(x/(1-x)). - _Vladimir Kruchinin_, Nov 28 2011
%F A007476 a(n) = A000994(n) + A000995(n). - _Peter Bala_, Jan 27 2015
%p A007476 a:= proc(n) option remember; `if`(n<2, 1,
%p A007476       add(a(j)*binomial(n-2, j), j=0..n-2))
%p A007476     end:
%p A007476 seq(a(n), n=0..31);  # _Alois P. Heinz_, Jul 29 2019
%t A007476 a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n-2, k] a[k], {k, 0, n-2}]; Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Aug 08 2012, after _Ralf Stephan_ *)
%o A007476 (PARI) a(n)=if(n<2, 1, sum(k=0, n-2, binomial(n-2, k)*a(k))) /* _Ralf Stephan_; corrected by Manuel Blum, May 22 2010 */
%Y A007476 Cf. A000994, A000995, A092920, A153859.
%Y A007476 Row sums of A246118.
%K A007476 nonn,eigen,nice
%O A007476 0,4
%A A007476 _N. J. A. Sloane_
%E A007476 Spelling correction by _Jason G. Wurtzel_, Aug 22 2010