cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A062752 Row sums of unsigned N(4) staircase array A062751.

Original entry on oeis.org

1, 15, 497, 22031, 1124849, 62379535, 3651676657, 222085764623, 13895337519601, 888654458770959, 57831897893972465, 3817410543738148367, 254970980461934291441, 17200148833928765494799
Offset: 0

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Formula

a(n)=N(4; n, -1) with the row polynomials N(4; n, x) defined in A062751.
a(n)=sum(((-1)^(n-j))*2^(3*j+1)*A002293(j), j=1..n)+(-1)^n, with A002293(j)= A062993(j+2, 2)= binomial(4*j, j)/(3*j+1).

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

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [Binomial(n+5,4) -1 : n in [0..50]]; // G. C. Greubel, Apr 22 2024
  • Maple
    [seq(binomial(n+5,4)-1,n=0..37)]; # Zerinvary Lajos, Nov 25 2006
  • Mathematica
    Binomial[5+Range[0,50],4] -1 (* G. C. Greubel, Apr 22 2024 *)
  • PARI
    { for (n=0, 1000, write("b063258.txt", n, " ", binomial(n + 5, 4) - 1) ) } \\ Harry J. Smith, Aug 19 2009
  • SageMath
    [binomial(n+5,4) -1 for n in range(51)] # G. C. Greubel, Apr 22 2024

Formula

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

Extensions

Simpler definition from Vladeta Jovovic, Jul 21 2003

A062750 Generalized Catalan array FS(4; n,r).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 1, 3, 6, 10, 14, 18, 22, 22, 22, 22, 1, 4, 10, 20, 34, 52, 74, 96, 118, 140, 140, 140, 140, 1, 5, 15, 35, 69, 121, 195, 291, 409, 549, 689, 829, 969, 969, 969, 969, 1, 6, 21, 56, 125
Offset: 0

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

In the Frey-Sellers reference this array is called {n over r}_{m-1}, with m=4.
The step width sequence of this staircase array is [1,3,3,3,....], i.e. the degree of the row polynomials is [0,3,6,9,...]= A008585.
The columns r=0..6 give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A000292 (tetrahedral), A063258, A027659, A062966.

Examples

			{1}; {1,1,1,1}; {1,2,3,4,4,4,4}; {1,3,6,10,14,18,22,22,22,22}; ...; N(4; 1,x)=(2-x)*(2-2*x+x^2).

Links

Formula

a(0, 0)=1, a(n, -1)=0, n >= 1; a(n, r)=0 if r>3*n; a(n, r)=a(n, r-1)+a(n-1, r) else.
G.f. for column r=3*k+j, k >= 0, j=1, 2, 3: (x^(k+1))*N(4; k, x)/(1-x)^(3*k+1+j), with the row polynomials N(4; k, x) of array A062751.

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

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

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.

Examples

			{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.

Crossrefs

A062991, A062746, A062751.

Formula

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.

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

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

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

Links

Crossrefs

Seventh column (r=6) of FS(4) staircase array A062750.
Partial sums of A027659.

Programs

  • Mathematica
    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 *)
  • PARI
    { 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

Formula

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

Extensions

Better description from Zerinvary Lajos, Dec 02 2005
Showing 1-5 of 5 results.

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