A062985 -id:A062985 - cp's OEIS frontend

A062990 Eighth column (r=7) of FS(5) staircase array A062985.

5, 30, 110, 315, 771, 1688, 3396, 6390, 11385, 19382, 31746, 50297, 77415, 116160, 170408, 245004, 345933, 480510, 657590, 887799, 1183787, 1560504, 2035500, 2629250, 3365505, 4271670, 5379210, 6724085, 8347215

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 7}_{4}, 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 (8,-28,56,-70,56,-28,8,-1).

Partial sums of A062989.

[seq((binomial(n+7,n)-binomial(n+2,n)),n=1..29)]; # Zerinvary Lajos, Jun 23 2006

Table[Binomial[n+7,n]-Binomial[n+2,n],{n,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{5,30,110,315,771,1688,3396,6390},30] (* Harvey P. Dale, Jun 09 2016 *)

{ for (n=0, 1000, m=n + 1; a=binomial(m + 7, m) - binomial(m + 2, m); write("b062990.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 15 2009

a(n) = A062985(n+2, 7) = (n+1)*(n+2)*(n+3)*(n^4 + 29*n^3 + 326*n^2 + 1744*n + 4200)/7!.
G.f.: N(5;1, x)/(1-x)^8 with N(5;1, x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x polynomial of second row of A062986.
a(n) = binomial(n+7,n) - binomial(n+2,n). - Zerinvary Lajos, Jun 23 2006

More terms from Zerinvary Lajos, Jun 23 2006

A062988 a(n) = binomial(n+6,5) - 1.

Original entry on oeis.org

5, 20, 55, 125, 251, 461, 791, 1286, 2001, 3002, 4367, 6187, 8567, 11627, 15503, 20348, 26333, 33648, 42503, 53129, 65779, 80729, 98279, 118754, 142505, 169910, 201375, 237335, 278255, 324631

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 5}_{4}, n >= 0.

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

Sixth column (r=5) of FS(5) staircase array A062985.
A column of triangle A014473.
Cf. A000096, A062748, A063258.

[Binomial(n+6,5) -1: n in [0..40]]; // G. C. Greubel, Apr 25 2024

[seq(binomial(n+6,5)-1, n=0..35)]; # Zerinvary Lajos, Nov 25 2006

Binomial[Range[6,45],5] -1 (* G. C. Greubel, Apr 25 2024 *)

{ for (n=0, 1000, write("b062988.txt", n, " ", binomial(n + 6, 5) - 1) ) } \\ Harry J. Smith, Aug 15 2009

[binomial(n+6,5) -1 for n in range(41)] # G. C. Greubel, Apr 25 2024

a(n) = A062985(n+2, 5).
a(n) = (n+1)*(n^4 + 19*n^3 + 136*n^2 + 444*n + 600)/5!.
G.f.: N(5;1, x)/(1-x)^6 with N(5;1, x)= 5 - 10*x + 10*x^2 - 5*x^3 + x^4 = (1-(1-x)^5)/x, polynomial of second row of A062986.
E.g.f.: (1/120)*(600 + 1800*x + 1200*x^2 + 300*x^3 + 30*x^4 + x^5)*exp(x). - G. C. Greubel, Apr 25 2024

A062989 a(n) = C(n+6, 6) - n - 1.

Original entry on oeis.org

0, 5, 25, 80, 205, 456, 917, 1708, 2994, 4995, 7997, 12364, 18551, 27118, 38745, 54248, 74596, 100929, 134577, 177080, 230209, 295988, 376717, 474996, 593750, 736255, 906165, 1107540, 1344875, 1623130, 1947761, 2324752, 2760648, 3262589, 3838345, 4496352

Offset: 0

Wolfdieter Lang, Jul 12 2001

In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{4}, 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(5) staircase array A062985.

Partial sums of A062988.

Table[Binomial[n+6,6]-n-1,{n,0,40}] (* OR *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,5,25,80,205,456,917},40] (* Harvey P. Dale, Aug 08 2013 *)

{ for (n=0, 1000, write("b062989.txt", n, " ", binomial(n + 6, 6) - n - 1) ) } \\ Harry J. Smith, Aug 15 2009

a(n) = A062985(n+2, 6) = (n+1)*(n+2)*(n^4 + 24*n^3 + 221*n^2 + 954*n + 1800)/6!.
G.f.: N(5;1, x)/(1-x)^7 with N(5;1, x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x polynomial of second row of A062986.
a(0)=0, a(1)=5, a(2)=25, a(3)=80, a(4)=205, a(5)=456, a(6)=917, 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, Aug 08 2013
D-finite with recurrence -n*a(n) +(n+6)*a(n-1) +5*n=0. - R. J. Mathar, Nov 22 2024

Simpler definition from Zerinvary Lajos, May 08 2006

A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

Original entry on oeis.org

1, 5, -10, 10, -5, 1, 35, -170, 415, -629, 630, -420, 180, -45, 5, 285, -2315, 9381, -24395, 44625, -59880, 60015, -45040, 25025, -10010, 2730, -455, 35, 2530, -29379, 169405, -633675, 1703700, -3467145, 5497640, -6903325

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column r=4*k+j, k >= 0, j=1,2,3,4, of the staircase array A062985(n,r) is N(5; k,x)*(x^(k+1))/(1-x)^(4*k+1+j) with N(5; k,x) := sum(a(k,p)*x^p,p=0..4*k).
The m=0 column gives A002294(k+1). The row sums give A000012 (powers of 1) and (unsigned) A062987.
The sequence of step width of this staircase array is [1,4,4,4,...], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.

			{1}; {5,-10,10,-5,1}; {35,-170,415,-629,630,-420,180,-45,5}; ...; N(5; 1,x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x.

A062991, A062746, A062751.

a(k, p) := [x^p]N(5; k, x) with N(5; k, x)=(N(5; k-1, x)- A002294(k)*(1-x)^(4*k+1))/x, N(5; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=0, .., (4*n-5); a(n, k)= ((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=(4*n-4), ..., 4*n; else 0.

A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).

Original entry on oeis.org

0, 4, 30, 250, 2245, 21221, 208129, 2098565, 21619910, 226593015, 2408424760, 25899375645, 281273231985, 3080585212120, 33986840371400, 377364606387005, 4213620859310140, 47284625533425750, 532996618440511710, 6032169040263819485, 68517222947120776290

Offset: 0

Andrei Asinowski, May 08 2020

A 4-Dyck path is a lattice path with steps U = (1, 4), 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) = 30 is the total number of down-steps after the last up-step in UddddUdddd, UdddUddddd, UddUdddddd, UdUddddddd, UUdddddddd (thus, 4 + 5 + 6 + 7 + 8).

Stefano Spezia, Table of n, a(n) for n = 0..900
Andrei Asinowski, Benjamin Hackl, and Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

First order differences of A002294. Cf. A062985.

Cf. A334682 (similar for 3-Dyck paths).

b:= proc(x, y) option remember; `if`(x=y, x,
     `if`(y+40, b(x-1, y-1), 0))
    end:
a:= n-> b(5*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n, (5*(5*n-4)*
      (5*n-3)*(5*n-2)*(5*n-1)*n*(2869*n^3+5354*n^2+3239*n+634)*
       a(n-1))/(8*(n-1)*(4*n+3)*(2*n+1)*(4*n+5)*(n+1)*
       (2869*n^3-3253*n^2+1138*n-120)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020

a[n_] := Binomial[5*n + 6, n + 1]/(5*n + 6) - Binomial[5*n + 1, n]/(5*n + 1); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = {binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1)} \\ Andrew Howroyd, May 08 2020

[binomial(5*(n + 1) + 1, n + 1)/(5*(n + 1) + 1) - binomial(5*n + 1, n)/(5*n + 1) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1).
a(n) = A062985(n+1, 4*n-1).
G.f.: ((1 - x)*HypergeometricPFQ([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023

A185915 Accumulation array of A185914, by antidiagonals.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 10, 9, 4, 1, 15, 16, 10, 4, 1, 21, 25, 19, 10, 4, 1, 28, 36, 31, 20, 10, 4, 1, 36, 49, 46, 34, 20, 10, 4, 1, 45, 64, 64, 52, 35, 20, 10, 4, 1, 55, 81, 85, 74, 55, 35, 20, 10, 4, 1, 66, 100, 109, 100, 80, 56, 35, 20, 10, 4, 1, 78, 121, 136, 130, 110, 83, 56, 35, 20, 10, 4, 1, 91, 144, 166, 164, 145, 116, 84, 56, 35, 20, 10, 4, 1, 105, 169, 199, 202, 185, 155, 119, 84, 56, 35, 20, 10, 4, 1

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185916 < A185914 < A185915 < ...

(See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
1....3....6....10....15....21....28
1....4....9....16....25....36....49
1....4....10...19....31....46....64
1....4....10...20....34....52....74
1....4....10...20....35....55....80
1....4....10...20....35....56....83
row 1: A000217 (triangular numbers)
row 2: A000290 (squares)
row 3: A005448 (centered triangular numbers)
row 4: A005893
row 5: A062985
Limit of rows: A000292 (tetrahedral numbers)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185914, A185916.

f[n_, 0] := 0; f[0, k_] := 0; f[n_, k_] := k - n + 1; f[n_, k_] := 0 /; k < n; s[n_, k_] := Sum[f[i, j], {i, 1, n}, {j, 1, k}]; Table[s[n - k + 1, k], {n, 50}, {k, n, 1, -1}] // Flatten

T(n,k) = C(k+2,3) if k<=n; T(n,k) = k*(k+2-n)/2 if k>n; k>=1, n>=1.

cp's OEIS Frontend

A062990 Eighth column (r=7) of FS(5) staircase array A062985.

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A062988 a(n) = binomial(n+6,5) - 1.

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A062989 a(n) = C(n+6, 6) - n - 1.

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A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

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A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

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A185915 Accumulation array of A185914, by antidiagonals.

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A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).