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 A001475 a(n) = a(n-1) + n * a(n-2), where a(1) = 1, a(2) = 2. (Formerly M1449 N0573) 7
 1, 2, 5, 13, 38, 116, 382, 1310, 4748, 17848, 70076, 284252, 1195240, 5174768, 23103368, 105899656, 498656912, 2404850720, 11879332048, 59976346448, 309442319456, 1628921941312, 8746095288800, 47840221880288, 266492604100288, 1510338372987776 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) is the number of set partitions of [n] in which the block containing 1 is of length <= 3 and all other blocks are of length <= 2. Example: a(4)=13 counts all 15 partitions of [4] except 1234 and 1/234. - David Callan, Jul 22 2008 Empirical: a(n) is the sum of the entries in the second-last row of the lower-triangular matrix of coefficients giving the expansion of degree-(n+1) complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018 REFERENCES J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 86 (divided by 2). N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS John Cerkan, Table of n, a(n) for n = 1..795 R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy] FORMULA E.g.f.: 1/2*(1+x)*exp(x+1/2*x^2) - 1/2. - Vladeta Jovovic, Nov 04 2003 Given e.g.f. y(x), then 0 = y'(x) * (1+x) - (y(x)+1/2) * (2+2*x+x^2) = 1 - y''(x) + y'(x)*(1 + x) + 2*y(x). - Michael Somos, Jan 23 2018 0 = +a(n)*(+a(n+1) +a(n+2) -a(n+3)) +a(n+1)*(-a(n+1) +a(n+2)) for all n>0. - Michael Somos, Jan 23 2018 a(n) ~ n^((n+1)/2) / (2^(3/2) * exp(n/2 - sqrt(n) + 1/4)) * (1 + 19/(24*sqrt(n))). - Vaclav Kotesovec, Apr 01 2018 EXAMPLE G.f. = x + 2*x + 5*x^2 + 13*x^3 + 38*x^4 + 116*x^5 + 382*x^6 + 1310*x^7 + ... - Michael Somos, Jan 23 2018 MAPLE a := proc(n) option remember: if n = 1 then 1 elif n = 2 then 2 elif  n >= 3 then procname(n-1) +n*procname(n-2) fi; end: seq(a(n), n = 1..100); # Muniru A Asiru, Jan 25 2018 MATHEMATICA RecurrenceTable[{a[1]==1, a[2]==2, a[n]==a[n-1]+n a[n-2]}, a, {n, 30}] (* Harvey P. Dale, Apr 21 2012 *) a[ n_] := With[{m = n + 1}, If[ m < 2, 0, Sum[(2 k - 1)!! Binomial[m, 2 k], {k, 0, m/2}] / 2]]; (* Michael Somos, Jan 23 2018 *) a[ n_] := With[{m = n + 1}, If[ m < 2, 0, HypergeometricU[ -m/2, 1/2, -1/2] / (-1/2)^(m/2) / 2]]; (* Michael Somos, Jan 23 2018 *) a[ n_] := With[{m = n + 1}, If[ m < 2, 0, HypergeometricPFQ[{-m/2, (1 - m)/2}, {}, 2] / 2]]; (* Michael Somos, Jan 23 2018 *) a[ n_] := If[ n < 1, 0, n! SeriesCoefficient[ Exp[ x + x^2/2] (1 + x)/2, {x, 0, n}]]; (* Michael Somos, Jan 23 2018 *) Fold[Append[#1, #1[[-1]] + #2 #1[[-2]]] &, {1, 2}, Range[3, 26]] (* Michael De Vlieger, Jan 23 2018 *) PROG (PARI) {a(n) = if( n<1, 0, n! * polcoeff( exp( x + x^2/2 + x * O(x^n)) * (1 + x) / 2, n))}; /* Michael Somos, Jan 23 2018 */ (GAP) a:=[1, 2];; for n in [3..10^2] do a[n] := a[n-1] + n*a[n-2]; od; a;  # Muniru A Asiru, Jan 25 2018 (MAGMA) I:=[1, 2]; [n le 2 select I[n] else Self(n-1)+n*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 31 2018 CROSSREFS Equals (1/2) A000085(n+1). Cf. A001189, A013989, A076276, A248475. Sequence in context: A064384 A148302 A149857 * A149858 A148303 A148304 Adjacent sequences:  A001472 A001473 A001474 * A001476 A001477 A001478 KEYWORD nonn AUTHOR EXTENSIONS More terms from Harvey P. Dale, Apr 21 2012 STATUS approved

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Last modified April 21 15:55 EDT 2021. Contains 343156 sequences. (Running on oeis4.)