A029887 -id:A029887 - 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

A045724 Convolution of Catalan numbers A000108 with A020918.

Original entry on oeis.org

1, 15, 142, 1083, 7266, 44758, 259356, 1435347, 7663898, 39761282, 201483204, 1001098462, 4891910100, 23565178380, 112118316088, 527674017411, 2459747256138, 11368724035210, 52145629874100, 237541552456362

Offset: 0

Wolfdieter Lang

Also convolution of A001700 with A038845; also convolution of A029887 with A000302 (powers of 4); also convolution of A042941 with A000984 (central binomial coefficients).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A000302, A000984, A001700, A020918, A029887, A042941.

[(Binomial(n+5,4)*Catalan(n+4) -5*4^(n+1)*Binomial(n+3,2))/10: n in [0..40]]; // G. C. Greubel, Jul 19 2024

Table[(Binomial[n+5,4]*CatalanNumber[n+4] -5*4^(n+1)*Binomial[n+3,2] )/10, {n,0,40}] (* G. C. Greubel, Jul 19 2024 *)

[(binomial(n+5,4)*catalan_number(n+4) - 5*4^(n+1)*binomial(n+3,2))/10 for n in range(41)] # G. C. Greubel, Jul 19 2024

a(n) = binomial(n+4, 3)*A000984(n+4)/(2*A000984(3)) - (n+3)*(n+2)*4^n, where A000984(n) = binomial(2*n, n),

G.f.: c(x)/(1-4*x)^(7/2) = (2 - c(x))/(1-4*x)^4, where c(x) = g.f. for Catalan numbers.

A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

Original entry on oeis.org

2, 22, 164, 1030, 5868, 31388, 160648, 795846, 3845020, 18211380, 84876152, 390331292, 1775032504, 7995075960, 35715205136, 158401506118, 698102372988, 3059470021316, 13341467466520, 57918065919924, 250419305769512, 1078769490401032, 4631680461623664, 19825379450255900, 84622558822506328, 360270317908904328, 1530148541536781488, 6484511936352543096, 27423786092731382000, 115756362341775227888

Offset: 2

N. J. A. Sloane

nonn

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. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Alois P. Heinz, Table of n, a(n) for n = 2..500
Richard P. Stanley, CATALAN ADDENDUM, version of Jul 19, 2008, p. 24. [From _Jonathan Vos Post_, Aug 16 2008]
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.
W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Column 3 of A269920.
Column 0 of A270407.
Cf. A000108, A029887.

[n*((n+1)*(n+2)*Catalan(n+1) - 3*4^n)/12: n in [2..30]]; // G. C. Greubel, Jul 18 2024

a[n_] := 1/12*(2^(n+1)*(2*n+1)!!/(n-1)!-3*4^n*n); Table[a[n], {n, 2, 31}] (* Jean-François Alcover, Mar 12 2014 *)

[n*(2*(2*n+1)*binomial(2*n,n) - 3*4^n)//12 for n in range(2,30)] # G. C. Greubel, Jul 18 2024

a(n) = 2 * A029887(n-2). - Ralf Stephan, Aug 17 2004
a(n) = 4^n*Gamma(n+3/2)/(3*sqrt(Pi)*Gamma(n)) - n*4^(n-1). - Mark van Hoeij, Jul 06 2010
From G. C. Greubel, Jul 18 2024: (Start)
a(n) = (n/12)*( (n+1)*(n+2)*Catalan(n+1) - 3*4^n ).
G.f.: x*(1 - sqrt(1 - 4*x))/(1-4*x)^(5/2).
E.g.f.: (x/3)*exp(2*x)*( - 3*exp(2*x) + 3*(1+2*x)*BesselI(0, 2*x) + (3+8*x)*BesselI(1, 2*x) + 2*x*BesselI(2, 2*x) ). (End)

More terms from Sean A. Irvine, Nov 14 2010

A046658 Triangle related to A001700 and A000302 (powers of 4).

Original entry on oeis.org

1, 3, 1, 10, 7, 1, 35, 38, 11, 1, 126, 187, 82, 15, 1, 462, 874, 515, 142, 19, 1, 1716, 3958, 2934, 1083, 218, 23, 1, 6435, 17548, 15694, 7266, 1955, 310, 27, 1, 24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1, 92378, 330818, 397923, 259356, 105102, 27866, 4867, 542, 35, 1

Offset: 1

Wolfdieter Lang

			Triangle begins as:
      1;
      3,     1;
     10,     7,     1;
     35,    38,    11,     1;
    126,   187,    82,    15,     1;
    462,   874,   515,   142,    19,    1;
   1716,  3958,  2934,  1083,   218,   23,   1;
   6435, 17548, 15694,  7266,  1955,  310,  27,  1;
  24310, 76627, 80324, 44758, 15086, 3195, 418, 31, 1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Column sequences for m=1..6: A001700, A000531, A029887, A045724, A045492, A045530.
Row sums: A046885.
Cf. A000302.

A046658:= func< n,k | Binomial(n,k)*(Binomial(n+1,2)*Catalan(n )/Catalan(k-1) -4^(n-k+1)*Binomial(k,2))/(n*(n-k+1)) >;
[A046658(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 28 2024

T[n_, k_]:= (1/2)*Binomial[n,k-1]*(Binomial[2*n,n]/Binomial[2*(k-1), k -1] - 4^(n-k+1)*(k-1)/n);
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)

def A046658(n,k): return (1/2)*binomial(n,k-1)*(binomial(2*n, n)/binomial(2*(k-1), k-1) - 4^(n-k+1)*(k-1)/n)
flatten([[A046658(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 28 2024

T(n, k) = (1/2)*binomial(n, k-1)*( binomial(2*n, n)/binomial(2*(k-1), k-1) - 4^(n-k+1)*(k-1)/n ), n >= k >= 1.

G.f. for column k: x*c(x)*((x/(1-4*x))^(k-1))/sqrt(1-4*x), where c(x) is the g.f. for Catalan numbers (A000108).

A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 10, 5, 1, 14, 35, 22, 7, 1, 42, 126, 93, 38, 9, 1, 132, 462, 386, 187, 58, 11, 1, 429, 1716, 1586, 874, 325, 82, 13, 1, 1430, 6435, 6476, 3958, 1686, 515, 110, 15, 1, 4862, 24310, 26333, 17548, 8330, 2934, 765, 142, 17, 1

Offset: 0

Philippe Deléham, Jan 25 2004

Read as a number triangle, this is the Riordan array (c(x),x/sqrt(1-4x)) where c(x) is the g.f. of A000108. - Paul Barry, May 16 2005

			row n=0 : 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.
row n=1 : 1, 3, 10, 35, 126, 462, 1716, 6435, ... see A001700.
row n=2 : 1, 5, 22, 93, 386, 1586, 6476, ... see A000346.
row n=3 : 1, 7, 38, 187, 874, 3958, 17548, ... see A000531.
row n=4 : 1, 9, 58, 325, 1686, 8330, 39796, ... see A018218.

Other rows : A029887, A042941, A045724, A042985, A045492. Columns : A000012, A005408. Row n is the convolution of the row (n-j) with A000984, A000302, A002457, A002697 (first term omitted), A002802, A038845, A020918, A038846, A020920 for j=1, 2, ..9 respectively.

T(n, k) = K_k(n)= Sum_{j>=0} A090285(k, j)*2^j*binomial(n, j). T(n, 1) = 2*n+1. T(n, 2) = 2*A028387(n).

Corrected by Alford Arnold, Oct 18 2006

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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A045724 Convolution of Catalan numbers A000108 with A020918.

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A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

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A046658 Triangle related to A001700 and A000302 (powers of 4).

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A090299 Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.

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