A370235 -id:A370235 - cp's OEIS frontend

A002802 a(n) = (2n+3)!/(6n!*(n+1)!).

1, 10, 70, 420, 2310, 12012, 60060, 291720, 1385670, 6466460, 29745716, 135207800, 608435100, 2714556600, 12021607800, 52895074320, 231415950150, 1007340018300, 4365140079300, 18839025605400, 81007810103220, 347176329013800, 1483389769422600

Offset: 0

N. J. A. Sloane

For n >= 1 a(n) is also the number of rooted bicolored unicellular maps of genus 1 on n+2 edges. - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 20 2001
a(n) is half the number of (n+2) X 2 Young tableaux with a three horizontal walls between the first and second column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021], A000984 for one horizontal wall, and A002457 for two. - Michael Wallner, Jan 31 2022
From Robert Coquereaux, Feb 12 2024: (Start)
Call B(p,g) the number of genus g partitions of a set with p elements (genus-dependent Bell number). Up to an appropriate shift the given sequence counts the genus 1 partitions of a set: we have a(n) = B(n+4,1), with a(0)= B(4,1)=1.
When shifted with an offset 4 (i.e., defining b(p)=a(p-4), which starts with 0,0,0,1,10,70, etc., and b(4)=1), the given sequence reads b(p) = (1/( 2^4 3 )) * (1/( (2 p - 1) (2 p - 3))) * (1/(p - 4)!) * (2p)!/p!. In this form it appears as a generalization of Catalan numbers (that indeed count the genus 0 partitions).
Call C[p, [alpha], g] the number of partitions of a set with p elements, of cyclic type [alpha], and of genus g (genus g Faa di Bruno coefficients of type [alpha]). Up to an appropriate shift the given sequence also counts the genus 1 partitions of p=2k into k parts of length 2, which is then called C[2k, [2^k], 1], and we have a(n) = C[2k, [2^k], 1] for k=n+2.
The two previous interpretations of this sequence, leading to a(n) = B(n+4, 1) and to a(n) = C[2(n+2), [2^(n+2)], 1] are not related in any obvious way. (End)

			G.f. = 1 + 10*x + 70*x^2 + 420*x^3 + 2310*x^4 + 12012*x^5 + 60060*x^6 + ...

C. Jordan, Calculus of Finite Differences. Röttig and Romwalter, Budapest, 1939; Chelsea, NY, 1965, p. 449.
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).

T. D. Noe, Table of n, a(n) for n = 0..200
Cyril Banderier and Michael Wallner, Young Tableaux with Periodic Walls: Counting with the Density Method, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See pp. 4, 12.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 9. See also arXiv:2305.01100, 2023. See pp. 9, 19.
R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
Alain Goupil and Gilles Schaeffer, Factoring N-Cycles and Counting Maps of Given Genus, Europ. J. Combinatorics (1998) 19 819-834.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. I, J. Comb. Theory B 13 (1972), 192-218 (Tab.1).
Liang Zhao and Fengyao Yan, Note on Total Positivity for a Class of Recursive Matrices, Journal of Integer Sequences, Vol. 19 (2016), Article 16.6.5.
Notes.

Cf. A035309, A000108 (for genus 0 maps), A046521 (third column).
Cf. A000984, A000911, A002457, A002697, A051133.
Column g=1 of A370235.

F:=Factorial;; List([0..25], n-> F(2*n+3)/(6*F(n)*F(n+1)) ); # G. C. Greubel, Jul 20 2019

F:=Factorial; [F(2*n+3)/(6*F(n)*F(n+1)): n in [0..25]]; // G. C. Greubel, Jul 20 2019

seq(simplify(4^n*hypergeom([-n,-3/2], [1], 1)),n=0..25); # Peter Luschny, Apr 26 2016

Table[(2*n+3)!/(6*n!*(n+1)!), {n, 0, 25}] (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)

{a(n) = if( n<0, 0, (2*n + 3)! / (6 * n! * (n+1)!))}; /* Michael Somos, Sep 16 2013 */

{a(n) = 2^(n+3) * polcoeff( pollegendre(n+4), n) / 3}; /* Michael Somos, Sep 16 2013 */

f=factorial; [f(2*n+3)/(6*f(n)*f(n+1)) for n in (0..25)] # G. C. Greubel, Jul 20 2019

G.f.: (1 - 4*x)^(-5/2) = 1F0(5/2;;4x).
Asymptotic expression for a(n) is a(n) ~ (n+2)^(3/2) * 4^(n+2) / (sqrt(Pi) * 48).
a(n) = Sum_{a+b+c+d+e=n} f(a)*f(b)*f(c)*f(d)*f(e) with f(n) = binomial(2n, n) = A000984(n). - Philippe Deléham, Jan 22 2004
a(n-1) = (1/4)*Sum_{k=1..n} k*(k+1)*binomial(2*k, k). - Benoit Cloitre, Mar 20 2004
a(n) = A051133(n+1)/3 = A000911(n)/6. - Zerinvary Lajos, Jun 02 2007
From Rui Duarte, Oct 08 2011: (Start)
Also convolution of A000984 with A002697, also convolution of A000302 with A002457.
a(n) = ((2n+3)(2n+1)/(3*1)) * binomial(2n, n).
a(n) = binomial(2n+4, 4) * binomial(2n, n) / binomial(n+2, 2).
a(n) = binomial(n+2, 2) * binomial(2n+4, n+2) / binomial(4, 2).
a(n) = binomial(2n+4, n+2) * (n+2)*(n+1) / 12. (End)
D-finite with recurrence: n*a(n) - 2*(2*n+3)*a(n-1) = 0. - R. J. Mathar, Jan 31 2014
a(n) = 4^n*hypergeom([-n,-3/2], [1], 1). - Peter Luschny, Apr 26 2016
Boas-Buck recurrence: a(n) = (10/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+2, 2). See a comment there. - Wolfdieter Lang, Aug 10 2017
a(n) = (-4)^n*binomial(-5/2, n). - Peter Luschny, Oct 23 2018
Sum_{n>=0} 1/a(n) = 12 - 2*sqrt(3)*Pi. - Amiram Eldar, Oct 13 2020
E.g.f.: (1/12) exp(2 x) x^2 BesselI[2, 2 x]. - Robert Coquereaux, Feb 12 2024

A297179 Number of genus-2 partitions of [n].

Original entry on oeis.org

1, 28, 399, 4179, 36498, 282282, 1999998, 13258674, 83417334, 503090588, 2929953026, 16569715890, 91386952020, 493234934220, 2612295374940, 13607257868820, 69841333755270, 353777814426960, 1770937330172010, 8770508370593970, 43015147164809820, 209104302965011740

Offset: 6

N. J. A. Sloane, Dec 27 2017

nonn

From Robert Coquereaux, Feb 12 2024: (Start)
Call B(n, g) the number of genus g partitions of a set with n elements (genus-dependent Bell number). Then a(n) = B(n, 2) with B(6, 2) = 1.
The entries of the triangle T(n, k) giving the number of genus 2 partitions of a set with n elements with k parts are known from R. Cori and G. Hetyei A297178.
Defining a(n) to be the sum over k of T(n,k) one shows that a(n) obeys the recurrence
a(n) = a(n-1) * (2*(-9 + 2*n) (-84 + n (88 + n*(-39 + 5*n)))) / ((-6 + n)*(-216 + n*(181 + n*(-54 + 5 n)))) with a(1) = a(2) = a(3) = a(4) = a(5) = 0 and a(6) = 1.
This determines a(n) for all n. One can solve the above recurrence and find an explicit formula, given below, for a(n) as a function of n. (End)

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 5.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 9.
Robert Cori and G. Hetyei, Counting partitions of a fixed genus, arXiv preprint arXiv:1710.09992 [math.CO], 2017.

Row sums of A297178.

Column g=2 of A370235.

a[n_] := (2^(n - 9) (88 n - 39 n^2 + 5 n^3 - 84) (2 n - 9)!!) / (45 (n - 6)!);
Table[a[n], {n, 6, 27}]  (* Peter Luschny, Feb 13 2024 *)

From Robert Coquereaux, Feb 12 2024: (Start)
a(n) = (1/(2^9*3^2*5)) * ((-84 + 88*n - 39*n^2 + 5*n^3) /((2*n - 1) * (2*n - 3) * (2*n - 5) * (2*n - 7))) * (1/(n - 6)!) * ((2*n)!/n!).
E.g.f.: (1/720) * exp(2*x) *(x^2*(-6 + 6*x - 9*x^2 + 5*x^3)*BesselI(0, 2*x) +  x*(6 - 6*x + 12*x^2 - 8*x^3 + 5*x^4)*BesselI(1, 2*x)). (End)

A370420 Number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). Flattened 3-dimensional array read by n, then by g:0..floor(n-1)/2, then by k:1..n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 0, 1, 0, 0, 1, 10, 20, 10, 1, 0, 5, 5, 0, 0, 1, 15, 50, 50, 15, 1, 0, 15, 40, 15, 0, 0, 0, 1, 0, 0, 0, 0, 1, 21, 105, 175, 105, 21, 1, 0, 35, 175, 175, 35, 0, 0, 0, 7, 21, 0, 0, 0, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 70, 560, 1050, 560, 70, 0, 0, 0, 28, 210, 161, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Robert Coquereaux, Feb 18 2024

Genus-dependent Stirling numbers of the second kind S2(n,k,g), 1 <= n, 1 <= k <= n, 0 <= g <= floor((n-1)/2). This is an infinite three-dimensional array. Its first 15 rows (n:1..15) are given by the table (see Links) taken from the article by Robert Coquereaux and Jean-Bernard Zuber (where a transpose of this table is given), see p. 32. These 15 rows determine 589 entries of the sequence (Data).
Example: the numbers S2(5,k,0), k=1..5, are {1,10,20,10,1} and appear on line 5, column 1; the numbers S2(5,k,1), k=1..5, are {0,5,5,0,0} and appear on line 5, column 2. Values of S2(n,k,g) for g > floor((n-1)/2) are equal to 0 and are not displayed.
Summing S2(n,k,g) over k gives genus-dependent Bell numbers B(n,g), A370235. Summing S2(n,k,g) over g gives S2(n,k), the Stirling numbers of the second kind A008277. Summing S2(n,k,g) over k and g gives the Bell numbers B(n), A000110. Example: S2(5,k,0) = 1, 10, 20, 10, 1 and S2(5,k,1) = 0, 5, 5, 0, 0 for k = 1..5; therefore S2(5,k) = 1, 15, 25, 10, 1, B(5,0) = 42, B(5,1) = 10, and B(5) = 52.

			For n:1..7, g:1..floor(n-1)/2, k:1..n. The 3-dimensional array begins:
  {1};
  {1,1};
  {1,3,1};
  {1,6,6,1},               {0,1,0,0};
  {1,10,20,10,1},          {0,5,5,0,0};
  {1,15,50,50,15,1},       {0,15,40,15,0,0},      {0,1,0,0,0,0};
  {1,21,105,175,105,21,1}, {0,35,175,175,35,0,0}, {0,7,21,0,0,0,0};

Robert Coquereaux, Rows n = 1..15 of array, flattened, terms 1..589
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 4, 5, 22.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 8, 9, 10, 32.
Robert Coquereaux, Table of S2(n,k,g) for n = 1..15
Robert Coquereaux, Mathematica program for the numbers S(n,k,g) and B(n,g)
Robert Cori and Gabor Hetyei, Counting partitions of a fixed genus, Electron. J. Combin. 25 (4) (2018), #P4.26.
Jean-Bernard Zuber, Counting partitions by genus. I. Genus 0 to 2, arXiv:2303.05875 [math.CO], 2023.
Jean-Bernard Zuber, Counting partitions by genus. I. Genus 0 to 2, Enumer. Comb. Appl. 4 (2) (2024) #S2R13.

Cf. A001263 (g=0), A370236 (g=1), A297178 (g=2).
Cf. A370235 (sum over k).
Cf. A000110, A008277.

Mathematica
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See Links
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No general formula is currently known. In the particular cases g=0, 1, 2, a formula is known: see Crossrefs.

A370237 Number of genus 3 partitions of the n-set.

Original entry on oeis.org

1, 94, 2620, 45430, 600655, 6633484, 64336844, 565256120

Offset: 8

Robert Coquereaux, Feb 12 2024

Call B(n, g) the number of genus g partitions of a set with n elements (genus-dependent Bell number). Then a(n) = B(n, 3) with B(8, 3) = 1.
a(8) = 1 through a(15) = 565256120 were explicitly determined by listing of partitions of an n-set and selecting those of genus 3.
The coefficients of the sixth-degree polynomial appearing in the numerator of the conjectured formula were determined by using experimental values for a(8) up to a(14); the term a(15) given by the formula agrees with the experimental value.
Using the conjectured formula for a(n) gives the following terms for n=16..20 :  4593034160, 35025118700, 253374008888, 1753071498620, 11675101781850. The E.g.f. given in the Formula section is obtained from the conjectured formula for a(n).

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 9. See also arXiv:2305.01100, 2023.

Column g=3 of A370235.

Cf. A000108, A002802, A297179, A001263, A370236, A297178.

Conjecture: a(n) = (1/(2^13 * 3^4 * 5 * 7)) * (35*n^6 - 819*n^5 + 7589*n^4 - 36009*n^3 + 93464*n^2 - 129060*n + 95040)/((2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)) * (1/(n-8)!) * (2*n)!/n!.

Conjecture: E.g.f.: (1/181440)*exp(2*x)*(x^2*(720 - 720*x + 1080*x^2 - 720*x^3 + 537*x^4 - 294*x^5 + 140*x^6)*BesselI(0, 2*x) + x*(-720 + 720*x - 1440*x^2 + 1080*x^3 - 1017*x^4 + 594*x^5 - 329*x^6 + 140*x^7)*BesselI(1, 2*x)).

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A002802 a(n) = (2*n+3)!/(6*n!*(n+1)!).

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A297179 Number of genus-2 partitions of [n].

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A370420 Number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). Flattened 3-dimensional array read by n, then by g:0..floor(n-1)/2, then by k:1..n.

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A370237 Number of genus 3 partitions of the n-set.

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A002802 a(n) = (2n+3)!/(6n!*(n+1)!).