A062750 -id:A062750 - cp's OEIS frontend

A063258 a(n) = binomial(n+5,4) - 1.

4, 14, 34, 69, 125, 209, 329, 494, 714, 1000, 1364, 1819, 2379, 3059, 3875, 4844, 5984, 7314, 8854, 10625, 12649, 14949, 17549, 20474, 23750, 27404, 31464, 35959, 40919, 46375, 52359, 58904, 66044, 73814, 82250, 91389, 101269, 111929, 123409, 135750

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 4}_{3}, n >= 0.
If X is an n-set and Y a fixed (n-4)-subset of X then a(n-5) is equal to the number of 4-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
For n>=5, a(n-5) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...01000 (the first n-5 zeros), or, the same, a(n-5) is up-down coefficient {n,8} (see comment in A060351). - Vladimir Shevelev, Feb 18 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000
Guillaume Aupy, Julien Herrmann. Periodicity in optimal hierarchical checkpointing schemes for adjoint computations. Optimization Methods and Software, Volume 32, 2017 - Issue 3. Preprint
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Fifth column (r=4) of FS(4) staircase array A062750.
A column of triangle A014473.
Cf. A000096, A062748, A062751.

[Binomial(n+5,4) -1 : n in [0..50]]; // G. C. Greubel, Apr 22 2024

[seq(binomial(n+5,4)-1,n=0..37)]; # Zerinvary Lajos, Nov 25 2006

Binomial[5+Range[0,50],4] -1 (* G. C. Greubel, Apr 22 2024 *)

{ for (n=0, 1000, write("b063258.txt", n, " ", binomial(n + 5, 4) - 1) ) } \\ Harry J. Smith, Aug 19 2009

[binomial(n+5,4) -1 for n in range(51)] # G. C. Greubel, Apr 22 2024

a(n) = A062750(n+2, 4) = (n+6)*(n+1)*(n^2 + 7*n + 16)/4!.
G.f.: (2-x)*(2-2*x+x^2)/(1-x)^5 = N(4;1, x)/(1-x)^5 with N(4;1, x)= 4 - 6*x + 4*x^2 - x^3, polynomial of second row of A062751.
E.g.f.: (1/24)*(96 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x). - G. C. Greubel, Apr 22 2024
a(n) = A000332(n+5)-1. - R. J. Mathar, Nov 22 2024

Simpler definition from Vladeta Jovovic, Jul 21 2003

A334682 a(n) is the total number of down-steps after the final up-step in all 3-Dyck paths of length 4n (n up-steps and 3n down-steps).

Original entry on oeis.org

0, 3, 18, 118, 829, 6115, 46736, 366912, 2941528, 23981628, 198224910, 1657364566, 13992405626, 119118427610, 1021399476720, 8813544248100, 76475285228304, 666865500290884, 5840843616021192, 51361847992315320, 453282040123194425, 4013440075484640675

Offset: 0

Andrei Asinowski, May 08 2020

nonn

A 3-Dyck path is a lattice path with steps U = (1, 3), d = (1, -1) that starts at (0,0), stays (weakly) above the x-axis, and ends at the x-axis.

			For n=2 the a(2)=18 is the total number of down-steps after the last up-step in UdddUddd, UddUdddd, UdUddddd, UUdddddd.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

First order differences of A002293.

Cf. A062750.

b:= proc(x, y) option remember; `if`(x=y, x,
     `if`(y+30, b(x-1, y-1), 0))
    end:
a:= n-> b(4*n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 3*n, (8*(4*n-1)*
      (2*n-1)*(4*n-3)*n*(229*n^2+303*n+98)*a(n-1))/
      (3*(n-1)*(3*n+2)*(3*n+4)*(n+1)*(229*n^2-155*n+24)))
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, May 09 2020

nmax = 21;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^4 + O[x]^(nmax + 2), nmax + 2];
CoefficientList[A[x], x] // Differences (* Jean-François Alcover, Aug 17 2020 *)

a(n) = {binomial(4*(n+1)+1, n+1)/(4*(n+1)+1) - binomial(4*n+1, n)/(4*n+1)} \\ Andrew Howroyd, May 08 2020

a(n) = binomial(4*(n+1)+1, n+1)/(4*(n+1)+1) - binomial(4*n+1, n)/(4*n+1).
a(n) = A062750(n+1, 3*n-1).
From Muhammed Sefa Saydam, Mar 01 2025: (Start)
Let F(n,k) = binomial((k+1)*n,n)/(k*n+1), then a(n) = F(n+1,3) - F(n,3).
Generally, F(y+1,x) - F(y,x) = Sum_{k=1..y} ( F(k+1,x) - T(x,k) + F(y-k,x) ) where T(n,k) = 2*(n+1)*binomial(n*k+k-1, k-1)/(n*k+1). (End)

A027659 a(n) = binomial(n+2,2) + binomial(n+3,3) + binomial(n+4,4) + binomial(n+5,5).

Original entry on oeis.org

4, 18, 52, 121, 246, 455, 784, 1278, 1992, 2992, 4356, 6175, 8554, 11613, 15488, 20332, 26316, 33630, 42484, 53109, 65758, 80707, 98256, 118730, 142480, 169884, 201348, 237307, 278226, 324601, 376960, 435864, 501908, 575722, 657972, 749361, 850630, 962559

Offset: 0

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A035343, A062750, A063422.

Partial sums of A063258.

[Binomial(n+6, 5) -(n+2): n in [0..60]]; // G. C. Greubel, Aug 01 2022

seq(1/120*(n+8)*(n+2)*(n+1)*(n^2+9*n+30), n=0..40);

Table[Sum[Binomial[n+i,i],{i,2,5}],{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20, -15,6,-1}, {4,18,52,121,246,455},30] (* Harvey P. Dale, Aug 18 2012 *)
Sum[(-1)^j*Binomial[4*j-2 + Range[0, 60], 4*j-3], {j,2}] (* G. C. Greubel, Aug 01 2022 *)

a(n)=(n+8)*(n+2)*(n+1)*(n^2+9*n+30)/120 \\ Charles R Greathouse IV, Oct 07 2015

[binomial(n+6, 5) -(n+2) for n in (0..60)] # G. C. Greubel, Aug 01 2022

a(n) = A035343(n+2, 5), n >= 0 (sixth column of quintinomial coefficients).
a(n) = A062750(n+2, 5), n >= 0 (sixth column).
G.f.: (x^2)*(2-x)*(2 - 2*x + x^2)/(1-x)^6. (For numerator polynomial see N5(5, x) = 4 - 6*x + 4*x^2 - x^3 from A063422.)
a(n) = binomial(n+6, 5) - binomial(n+2, 1). - Zerinvary Lajos, May 08 2006
a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6), with a(0)=4, a(1)=18, a(2)=52, a(3)=121, a(4)=246, a(5)=455. - Harvey P. Dale, Aug 18 2012
From G. C. Greubel, Aug 01 2022: (Start)
a(n) = Sum_{j=0..3} binomial(n+j+2, j+2).
E.g.f.: (1/120)*(480 +1680*x +1200*x^2 +300*x^3 +30*x^4 +x^5)*exp(x). (End)

A062751 Coefficient array for certain polynomials N(4; k,x) (rising powers in x).

Original entry on oeis.org

1, 4, -6, 4, -1, 22, -80, 139, -140, 84, -28, 4, 140, -851, 2500, -4536, 5516, -4616, 2640, -990, 220, -22, 969, -8420, 35504, -94584, 175564, -237600, 239250, -179960, 100078, -40040, 10920, -1820, 140, 7084, -80776, 448056

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column r=3*k+j, k >= 0, j=1,2,3, of the staircase array A062750(n,r) is N(4; k,x)*(x^(k+1))/(1-x)^(3*k+1+j) with N(4; k,x) := sum(a(k,p)*x^p,p=0..3*k).
The m=0 column gives: A002293(n+1). The row sums give A000012 (powers of 1) and (unsigned) A062752.
The sequence of step width of this staircase array is [1,3,3,3,...], i.e. the degree of the row polynomials is [0,3,6,9,...]= A008585.

			{1}; {4,-6,4,-1}; {22,-80,139,-140,84,-28,4}; ...; N(4; 1,x)= 4-6*x+4*x^2-x^3 =(2-x)*(2-2*x+x^2).

a(k, p) := [x^p]N(4; k, x) with N(4; k, x)=(N(4; k-1, x)-A002293(k)*(1-x)^(3*k+1))/x, N(4; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(3*n+1, k+1)*binomial(4*n+1, n)/(4*n+1) if k=0, .., (3*n-4); a(n, k)= ((-1)^k)*binomial(3*n+1, k+1)*binomial(4*n+1, n)/(4*n+1) if k=(3*n-3), ..., 3*n; else 0.

A062966 a(n) = C(3+n, n) + C(4+n, n) + C(5+n, n) + C(6+n, n).

Original entry on oeis.org

4, 22, 74, 195, 441, 896, 1680, 2958, 4950, 7942, 12298, 18473, 27027, 38640, 54128, 74460, 100776, 134406, 176890, 229999, 295757, 376464, 474720, 593450, 735930, 905814, 1107162, 1344469, 1622695, 1947296

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{3}, n >= 0.

Harry J. Smith, Table of n, a(n) for n = 0..1000
D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Seventh column (r=6) of FS(4) staircase array A062750.

Partial sums of A027659.

Table[Sum[Binomial[i+n,n],{i,3,6}],{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{4,22,74,195,441,896,1680},30] (* Harvey P. Dale, May 02 2012 *)

{ for (n=0, 1000, a=binomial(3 + n, n) + binomial(4 + n, n) + binomial(5 + n, n) + binomial(6 + n, n); write("b062966.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 14 2009

a(n) = A062750(n+2, 6) = (n+10)*(n+3)*(n+2)*(n+1)*(n^2+11*n+48)/6!.
G.f.: = (x-2)*(x^2-2*x+2)/(x-1)^7 = N(4;1, x)/(1-x)^7 with N(4;1, x) = 4-6*x+4*x^2-x^3, polynomial of second row of A062751.
a(0)=4, a(1)=22, a(2)=74, a(3)=195, a(4)=441, a(5)=896, a(6)=1680, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - Harvey P. Dale, May 02 2012

Better description from Zerinvary Lajos, Dec 02 2005

cp's OEIS Frontend

A063258 a(n) = binomial(n+5,4) - 1.

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A334682 a(n) is the total number of down-steps after the final up-step in all 3-Dyck paths of length 4*n (n up-steps and 3*n down-steps).

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A027659 a(n) = binomial(n+2,2) + binomial(n+3,3) + binomial(n+4,4) + binomial(n+5,5).

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Maple

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A062751 Coefficient array for certain polynomials N(4; k,x) (rising powers in x).

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Author

Keywords

Comments

Examples

Formula

A062966 a(n) = C(3+n, n) + C(4+n, n) + C(5+n, n) + C(6+n, n).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A334682 a(n) is the total number of down-steps after the final up-step in all 3-Dyck paths of length 4n (n up-steps and 3n down-steps).