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.

A062987 Row sums of unsigned N(5) staircase array A062986.

Original entry on oeis.org

1, 31, 2529, 284191, 37071329, 5268723231, 791682591201, 123697944483359, 19894672175770081, 3271817054307112479, 547678880100062177761, 93006445178165754746399, 15983911852747899752786401
Offset: 0

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Crossrefs

Cf. A002294, A062986, A062993.

Formula

a(n) = N(5; n, -1) with polynomials N(5; n, x) defined in A062986.
a(n) = Sum(((-1)^(n-j))*2^(4*j+1)*A002294(j), j=1..n)+(-1)^n, with A002294(j) = A062993(j+3, 3) = binomial(5*j, j)/(4*j+1).

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

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [Binomial(n+6,5) -1: n in [0..40]]; // G. C. Greubel, Apr 25 2024
  • Maple
    [seq(binomial(n+6,5)-1, n=0..35)]; # Zerinvary Lajos, Nov 25 2006
  • Mathematica
    Binomial[Range[6,45],5] -1 (* G. C. Greubel, Apr 25 2024 *)
  • PARI
    { for (n=0, 1000, write("b062988.txt", n, " ", binomial(n + 6, 5) - 1) ) } \\ Harry J. Smith, Aug 15 2009
  • SageMath
    [binomial(n+6,5) -1 for n in range(41)] # G. C. Greubel, Apr 25 2024

Formula

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

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{4}, n >= 0.

Links

Crossrefs

Seventh column (r=6) of FS(5) staircase array A062985.
Partial sums of A062988.

Programs

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

Formula

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

Extensions

Simpler definition from Zerinvary Lajos, May 08 2006

A062985 Generalized Catalan array FS(5; n,r).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 5, 1, 3, 6, 10, 15, 20, 25, 30, 35, 35, 35, 35, 35, 1, 4, 10, 20, 35, 55, 80, 110, 145, 180, 215, 250, 285, 285, 285, 285, 285, 1, 5, 15, 35, 70, 125, 205, 315, 460, 640, 855, 1105
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=5.
The step width sequence of this staircase array is [1,4,4,4,....], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.
The columns r=0..7 (without leading zeros) give A000012 (powers of 1), A000027 (natural), A000217 (triangular), A000292 (tetrahedral), A000332(4+n), A062988-A062990.

Examples

			{1}; {1,1,1,1,1}; {1,2,3,4,5,5,5,5,5}; ...; N(5; 1,x)=5-10*x+10*x^2-5*x^3+x^4.

Links

Formula

a(0, 0)=1, a(n, -1)=0, n >= 1; a(n, r)=0 if r>4*n; a(n, r)=a(n, r-1)+a(n-1, r) else.
G.f. for column r=4*k+j, k >= 0, j=1, 2, 3, 4: (x^(k+1))*N(5; k, x)/(1-x)^(4*k+1+j), with row polynomials of array A062986.

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

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Jul 12 2001

Keywords

Comments

In the Frey-Sellers reference this sequence is called {(n+2) over 7}_{4}, n >= 0.

Links

Crossrefs

Partial sums of A062989.

Programs

  • Maple
    [seq((binomial(n+7,n)-binomial(n+2,n)),n=1..29)]; # Zerinvary Lajos, Jun 23 2006
  • Mathematica
    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 *)
  • PARI
    { 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

Formula

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

Extensions

More terms from Zerinvary Lajos, Jun 23 2006
Showing 1-5 of 5 results.

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