A002802 -id:A002802 - cp's OEIS frontend

A258431 Sum over all peaks of Dyck paths of semilength n of the arithmetic mean of the x and y coordinates.

0, 1, 5, 23, 102, 443, 1898, 8054, 33932, 142163, 592962, 2464226, 10209620, 42190558, 173962532, 715908428, 2941192472, 12065310083, 49428043442, 202249741418, 826671597572, 3375609654698, 13771567556012, 56138319705908, 228669994187432, 930803778591278

Offset: 0

Alois P. Heinz, May 29 2015

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U = (1,1) and D = (1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Wikipedia, Average, Arithmetic mean
Wikipedia, Lattice path

Cf. A000302, A000346, A000531, A002457, A002697, A002802, A029887.

A258431:= func< n | n eq 0 select 0 else (4^(n-1) + Factorial(2*n-1)/Factorial(n-1)^2)/2 >;
[A258431(n): n in [0..40]]; // G. C. Greubel, Mar 18 2023

a:= proc(n) option remember; `if`(n<3, [0, 1, 5][n+1],
       ((8*n-10)*a(n-1)-(16*n-24)*a(n-2))/(n-1))
    end:
seq(a(n), n=0..30);

a[0]=0; a[1]=1; a[2]=5;
a[n_]:= a[n]= (2*(4*n-5)*a[n-1] - 8*(2*n-3)*a[n-2])/(n-1);
Table[a[n], {n,0,30}] (* Jean-François Alcover, May 31 2018, from Maple *)

def A258431(n): return 0 if (n==0) else (4^(n-1) + factorial(2*n-1)/factorial(n-1)^2)/2
[A258431(n) for n in range(41)] # G. C. Greubel, Mar 18 2023

G.f.: x*(1 + sqrt(1-4*x))/(2*sqrt(1-4*x)^3).
a(n) = (2*(4*n-5)*a(n-1) - 8*(2*n-3)*a(n-2))/(n-1) for n>2, a(0)=0, a(1)=1, a(2)=5.
a(n) = (4^(n-1) + (2*n-1)!/(n-1)!^2)/2 for n>0, a(0) = 0.
a(n) = (A000302(n-1) + A002457(n-1))/2 for n>0, a(0) = 0.
a(n) = (1/2)*binomial(2*n,n)*( 1 + 2*(n-1)/(n+1) + 3*(n-1)*(n-2)/((n+1)*(n+2)) + 4*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + 5*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + ...) for n >= 1. - Peter Bala, Feb 17 2022

A000911 a(n) = (2n+3)! /( n! * (n+1)! ).

Original entry on oeis.org

6, 60, 420, 2520, 13860, 72072, 360360, 1750320, 8314020, 38798760, 178474296, 811246800, 3650610600, 16287339600, 72129646800, 317370445920, 1388495700900, 6044040109800, 26190840475800, 113034153632400, 486046860619320, 2083057974082800, 8900338616535600

Offset: 0

N. J. A. Sloane

nonn

			6 + 60*x + 420*x^2 + 2520*x^3 + 13860*x^4 + 72072*x^5 + 360360*x^6 + ...

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99.

T. D. Noe, Table of n, a(n) for n = 0..200

Cf. A000217, A000984, A001801, A002802, A051133, A086466, A093602.

seq(binomial(2*n,n)*binomial(n,(n-2)), n=2..21); # Zerinvary Lajos, May 10 2007

Table[(2 n + 3)!/(n!*(n + 1)!), {n, 0, 20}] (* T. D. Noe, Jun 20 2012 *)

a(n) = 2^(n+4)*polcoeff(pollegendre(n+4),n) /* Ralf Stephan */

a(n) = 2 * A051133(n+1).
a(n) = A000984(n+1)*A000217(n). - Zerinvary Lajos, May 10 2007
a(n) = 6 * A002802(n). - Zerinvary Lajos, Jun 02 2007
n*a(n) - 2*(2*n+3)*a(n-1) = 0. - R. J. Mathar, Jun 07 2013
G.f.: 6*(1+10*x/( G(0)- 10*x)), where G(k)= 2*x*(2*k+5) + k + 1 - 2*x*(k+1)*(2*k+7)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 14 2013
Sum_{n>=0} (-1)^n/a(n) = 5*A086466-2 = 2*log(phi)*sqrt(5)-2 = 0.1520447... - Jean-François Alcover, Apr 22 2014
From Ilya Gutkovskiy, Jan 31 2017: (Start)
G.f.: 6/(1 - 4*x)^(5/2).
a(n) ~ 2^(2*n+3)*n^(3/2)/sqrt(Pi). (End)
Sum_{n>=0} 1/a(n) = 2 - Pi/sqrt(3) = 2 - A093602. - Amiram Eldar, Oct 13 2020

A370235 Table read by rows. Number of set partitions of [n] with respect to genus g.

Original entry on oeis.org

1, 1, 2, 5, 14, 1, 42, 10, 132, 70, 1, 429, 420, 28, 1430, 2310, 399, 1, 4862, 12012, 4179, 94, 16796, 60060, 36498, 2620, 1, 58786, 291720, 282282, 45430, 352, 208012, 1385670, 1999998, 600655, 19261, 1, 742900, 6466460, 13258674, 6633484, 541541, 1378

Offset: 0

Peter Luschny, Feb 15 2024

The table shows the number of partitions of [n] = {1, 2, 3, ..., n} with genus g.
The set of noncrossing partitions is exactly the set of genus zero partitions. The numbers corresponding to this case are the Catalan numbers.
This is essentially table 2.1 in Martha Yip's thesis (p. 12).
From Robert Coquereaux, Feb 16 2024: (Start)
The two-dimensional array is called triangle of genus-dependent Bell numbers B(n, g); if n >= 1, n even, nonzero values are obtained for 0 <= g <= floor((n-1)/2); if n >= 1, odd, nonzero values are obtained for 0 <= g < (n-1)/2.
The two-dimensional array B(n, g) can be obtained from a three-dimensional array S2(n, k, g), by summation over the number k of blocks. The numbers S2(n, k, g) are genus-dependent Stirling numbers of the second kind. They give the number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). The numbers S2(n, k, g) are discussed in A370420.
(End)

			[n\g]     0        1        2      3      4     5
-------------------------------------------------
[ 0]      1;
[ 1]      1;
[ 2]      2;
[ 3]      5;
[ 4]     14,       1;
[ 5]     42,      10;
[ 6]    132,      70,        1;
[ 7]    429,     420,       28;
[ 8]   1430,    2310,      399,       1;
[ 9]   4862,   12012,     4179,      94;
[10]  16796,   60060,    36498,    2620,      1;
[11]  58786,  291720,   282282,   45430,    352;
[12] 208012, 1385670,  1999998,  600655,  19261,    1;
[13] 742900, 6466460, 13258674, 6633484, 541541, 1378;

Robert Coquereaux, Table of n, a(n) for n = 0..57 (rows 0..15)
Martha Yip, Genus one partitions, Master Thesis, University of Waterloo, 2006. [Typos in Table 2.1 in positions T(8, 0) and T(10, 0)].
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. Also in JIS, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 8, 9, 10, 32.

Columns: A000108 (g=0), A002802 (g=1), A297179 (g=2), A370237 (g=3).
Cf. A000110 (row sums), A177267 (permutations by genus).
Cf. A001263, A370236, A297178.
Cf. A370420 (S2(n,k,g)).

A002738 Coefficients for extrapolation.

Original entry on oeis.org

3, 60, 630, 5040, 34650, 216216, 1261260, 7001280, 37413090, 193993800, 981608628, 4867480800, 23728968900, 114011377200, 540972351000, 2538963567360, 11802213457650, 54396360988200, 248812984520100, 1130341536324000, 5103492036502860, 22913637714910800

Offset: 0

N. J. A. Sloane

nonn

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-2 of B equals a(n-3). - T. D. Noe, May 01 2011

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
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).

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

Cf. A002802, A007744, A020918.

A diagonal of A331431.

[3*Binomial(2*n+3,n)*Binomial(n+3,3): n in [0..30]]; // G. C. Greubel, Mar 21 2022

Table[Total[Inverse[HilbertMatrix[n]][[n - 2]]], {n, 3, 25}] (* T. D. Noe, May 02 2011 *)

[3*binomial(2*n+3,3)*binomial(2*n,n) for n in (0..30)] # G. C. Greubel, Mar 21 2022

From Alois P. Heinz, May 02 2011: (Start)
a(n) = 3*binomial(2*n+3,n)*binomial(n+3,n).
G.f.: 3*(1 + 6*x)/(1-4*x)^(7/2). (End)
a(n) = binomial(2*n+3,n)*(n^3 + 6*n^2 + 11*n+6)/2. - Charles R Greathouse IV, May 02 2011
a(n) = 3*A007744(n). - R. J. Mathar, Jan 21 2020
a(n) = (3/2)*( 5*A020918(n) - 3*A002802(n)). - G. C. Greubel, Mar 21 2022

Extended by T. D. Noe, May 01 2011

A370236 Triangle read by rows: T(n, k) is the number of partitions of genus 1 and k parts of the n-set (n >= 4, 2 <= k <= n-2).

Original entry on oeis.org

1, 5, 5, 15, 40, 15, 35, 175, 175, 35, 70, 560, 1050, 560, 70, 126, 1470, 4410, 4410, 1470, 126, 210, 3360, 14700, 23520, 14700, 3360, 210, 330, 6930, 41580, 97020, 97020, 41580, 6930, 330, 495, 13200, 103950, 332640, 485100, 332640, 103950, 13200, 495

Offset: 4

Robert Coquereaux, Feb 12 2024

The formula given below was conjectured by Martha Yip and proved by Robert Cori and Gábor Hetyei.
More generally one may consider genus-dependent Stirling numbers S(n, k, g) that count the partitions of genus g and k parts of the n-set.
Then T(n, k) = S(n, k, 1). See Robert Coquereaux and Jean-Bernard Zuber.

			Triangle begins (see Table 3.1 in Yip's thesis):
    1;
    5,    5;
   15,   40,   15;
   35,  175,  175,   35;
   70,  560, 1050,  560,   70;
  126, 1470, 4410, 4410, 1470, 126;

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.
Robert Cori and Gábor Hetyei, Counting genus one partitions and permutations, arXiv:1306.4628 [math.CO], 2013.
Robert Cori and Gábor Hetyei, Counting genus one partitions and permutations, Sémin. Lothar. Comb. 70, B70e, 30 p. (2014).
Martha Yip, Genus one partitions, Master Thesis, University of Waterloo, 2006.

Row sums are A002802.

Cf. A000332, A297178 (genus 2).

T[n_,k_] := (1/6) Binomial[n, 2] Binomial[n-2, k] Binomial[n-2, k-2];
Table[T[n,k],{n,4,12},{k,2,n-2}]//Flatten (* Stefano Spezia, Feb 14 2024 *)

T(n, k) = (1/6)*binomial(n, 2)*binomial(n-2, k)*binomial(n-2, k-2).

A382515 Expansion of 1/(1 - x/(1 - 4*x)^(5/2)).

Original entry on oeis.org

1, 1, 11, 91, 691, 5101, 37323, 272405, 1987047, 14493479, 105718071, 771148119, 5625136651, 41032826127, 299316769887, 2183389173811, 15926906427179, 116180104751925, 847485191674867, 6182049517420133, 45095462188117951, 328952511222499589, 2399570809473795931

Offset: 0

Seiichi Manyama, Mar 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A026671, A382514.

Cf. A002802.

Table[Sum[4^(n-k)*Binomial[n+3*k/2-1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Mar 30 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*binomial(n+3*k/2-1, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n+3*k/2-1,n-k).

D-finite with recurrence (-n+1)*a(n) +2*(12*n-19)*a(n-1) +(-239*n+519)*a(n-2) +2*(638*n-1751)*a(n-3) +1280*(-3*n+10)*a(n-4) +512*(12*n-47)*a(n-5) +2048*(-2*n+9)*a(n-6)=0. - R. J. Mathar, Mar 31 2025

A387341 Expansion of 1/(1 - 6*x + x^2)^(5/2).

Original entry on oeis.org

1, 15, 155, 1365, 10990, 83538, 610050, 4325310, 29979015, 204086025, 1369220853, 9075850875, 59550467340, 387359772660, 2500864350900, 16040872988748, 102298452571965, 649077104453715, 4099652984281855, 25788295829930865, 161619907171129946, 1009512779437342950

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A002802, A387313, A387343.

Cf. A277660.

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 6*x + x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/(1-6*x+x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/(1-6*x+x^2)^(5/2))

n*a(n) = 3*(2*n+3)*a(n-1) - (n+3)*a(n-2) for n > 1.
a(n) = (binomial(n+4,2)/6) * A387337(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/6)^(n-2*k) * binomial(-5/2,k) * binomial(k,n-k).
a(n) = A277660(n+2)/2.
a(n) ~ n^(3/2) * (1 + sqrt(2))^(2*n+5) / (3*2^(17/4)*sqrt(Pi)). - Vaclav Kotesovec, Aug 27 2025

A041001 Convolution of A000108(n+1), n >= 0, (Catalan numbers) with A038845 (3-fold convolution of powers of 4).

Original entry on oeis.org

1, 14, 125, 906, 5810, 34364, 191901, 1026610, 5312230, 26767940, 131990066, 639210404, 3048892740, 14354652152, 66828135005, 308078809794, 1408022619806, 6385966846580, 28765327498278, 128777533131500

Offset: 0

Wolfdieter Lang

Also convolution of A038836 with A000984 (central binomial coefficients); also convolution of A001791(n+1), n >= 0, with A002802; also convolution of A008549(n+1), n >= 0, with A002697; also convolution of A029760 with A002457; also convolution of A038806(n+1), n >= 0, with A000302 (powers of 4).

a(n) = (n+3)*(3*(n+6)*2^(2*n+3)-(n+4)*binomial(2*n+7, n+3))/12; G.f. (c(x)^2)/(1-4*x)^3, where c(x) = g.f. for Catalan numbers.

A323223 a(n) = [x^n] x/((1 - x)(1 - 4x)^(5/2)).

Original entry on oeis.org

0, 1, 11, 81, 501, 2811, 14823, 74883, 366603, 1752273, 8218733, 37964449, 173172249, 781607349, 3496163949, 15517771749, 68412846069, 299828796219, 1307168814519, 5672308893819, 24511334499219, 105519144602439, 452695473616239, 1936085243038839, 8256615564926439

Offset: 0

Peter Luschny, Jan 26 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1653

Row 5 of A323222.

Cf. A002802.

A323223List := proc(len) local ogf, ser; ogf := (1 - 4*x)^(-5/2)*x/(1 - x);
ser := series(ogf, x, (n+1)*len+1); seq(coeff(ser, x, j), j=0..len) end:
A323223List(24);
# Alternative:
a := proc(n) option remember; `if`(n<2,n,((5*n+1)*a(n-1)-(4*n+2)*a(n-2))/(n-1)) end: seq(a(n), n=0..24);

a(n) = ((5*n + 1)*a(n-1) - (4*n + 2)*a(n-2))/(n - 1) for n >= 2.
a(n) = -(-4)^n*binomial(-5/2, n)*hypergeom([1, n+5/2], [n+1], 4) - i*sqrt(3)/27.
a(n) ~ 2^(2*n+2) * n^(3/2) / (9*sqrt(Pi)). - Vaclav Kotesovec, Jan 29 2019
a(n+1) - a(n) = A002802(n). - Seiichi Manyama, Jan 29 2019

A387343 Expansion of 1/(1 - 8x + 4x^2)^(5/2).

Original entry on oeis.org

1, 20, 270, 3080, 31990, 312984, 2937900, 26751120, 237977190, 2078447800, 17884238372, 152002796400, 1278603975740, 10660760170480, 88213513627800, 725107271106336, 5925674432448390, 48175954959638520, 389871795632108020, 3142078444590396080, 25228464363569709396

Offset: 0

Seiichi Manyama, Aug 27 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A069835, A331515, A387344.
Cf. A002802, A387313, A387341.
Cf. A387339.

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 8*x + 4*x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

CoefficientList[Series[1/(1-8*x+4*x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(5/2))

n*a(n) = 4*(2*n+3)*a(n-1) - 4*(n+3)*a(n-2) for n > 1.
a(n) = (binomial(n+4,2)/6) * A387339(n).
a(n) = (-1)^n * Sum_{k=0..n} (1/2)^(n-4*k) * binomial(-5/2,k) * binomial(k,n-k).

cp's OEIS Frontend

A258431 Sum over all peaks of Dyck paths of semilength n of the arithmetic mean of the x and y coordinates.

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A000911 a(n) = (2n+3)! /( n! * (n+1)! ).

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A370235 Table read by rows. Number of set partitions of [n] with respect to genus g.

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A002738 Coefficients for extrapolation.

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A370236 Triangle read by rows: T(n, k) is the number of partitions of genus 1 and k parts of the n-set (n >= 4, 2 <= k <= n-2).

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A382515 Expansion of 1/(1 - x/(1 - 4*x)^(5/2)).

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A387341 Expansion of 1/(1 - 6*x + x^2)^(5/2).

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A323223 a(n) = [x^n] x/((1 - x)(1 - 4x)^(5/2)).

A387343 Expansion of 1/(1 - 8x + 4x^2)^(5/2).