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.

A000575 Tenth column of quintinomial coefficients.

Original entry on oeis.org

10, 80, 365, 1246, 3535, 8800, 19855, 41470, 81367, 151580, 270270, 464100, 771290, 1245488, 1960610, 3016820, 4547840, 6729800, 9791859, 14028850, 19816225, 27627600, 38055225, 51833730, 69867525, 93262260, 123360780, 161784040, 210477476, 271763360
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In the Carlitz et al. reference a(n)= Q_{5,n+2}(2), n >= 0, with a(n)=binomial(11+n,n+2)-(n+3)*binomial(n+6,n+2), (eq.(3.3), p. 356, with n=5, m->n+2,r=2). Q_{5,m}(2) is the number of sequences (i_1,i_2,...,i_m) with i_s, s=1,...,m, from {1,2,3,4,5} (repetitions allowed), with exactly 2 increases between successive elements (first position is counted as an increase).

References

  • 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).

Links

Programs

  • Mathematica
    CoefficientList[Series[(10-20*x+15*x^2-4*x^3)/(1-x)^10,{x,0,50}],x](* Vincenzo Librandi, Mar 28 2012 *)
  • PARI
    a(n) = polcoeff((1+x+x^2+x^3+x^4)^(n+3), 9); \\ Joerg Arndt, Aug 04 2015

Formula

a(n) = A035343(n+3, 9) = binomial(n+6, 6)*(n^3+42*n^2+677*n+5040)/(9!/6!).
G.f.: (10-20*x+15*x^2-4*x^3)/(1-x)^10; numerator polynomial is N5(9, x) from the array A063422.
a(n) = 10*C(n+3,3) + 40*C(n+3,4) + 65*C(n+3,5) + 56*C(n+3,6) + 28*C(n+3,7) + 8*C(n+3,8) + C(n+3,9) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
a(n) = Sum_{k=1..10} (-1)^k * binomial(10,k) * a(n-k), a(0)=10. - G. C. Greubel, Aug 03 2015
a(n) = [x^9] (1+x+x^2+x^3+x^4)^(n+3). - Joerg Arndt, Aug 04 2015

Extensions

Comments and more terms from Wolfdieter Lang, Aug 29 2001
More terms from Sean A. Irvine, Nov 24 2010

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

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035343, A062750, A063422.
Partial sums of A063258.

Programs

  • Magma
    [Binomial(n+6, 5) -(n+2): n in [0..60]]; // G. C. Greubel, Aug 01 2022
  • Maple
    seq(1/120*(n+8)*(n+2)*(n+1)*(n^2+9*n+30), n=0..40);
  • Mathematica
    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 *)
  • PARI
    a(n)=(n+8)*(n+2)*(n+1)*(n^2+9*n+30)/120 \\ Charles R Greathouse IV, Oct 07 2015
  • SageMath
    [binomial(n+6, 5) -(n+2) for n in (0..60)] # G. C. Greubel, Aug 01 2022

Formula

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)

A064056 Seventh column of quintinomial coefficients.

Original entry on oeis.org

3, 19, 68, 185, 426, 875, 1652, 2922, 4905, 7887, 12232, 18395, 26936, 38535, 54008, 74324, 100623, 134235, 176700, 229789, 295526, 376211, 474444, 593150, 735605, 905463, 1106784, 1344063, 1622260, 1946831
Offset: 0

Views

Author

Wolfdieter Lang, Aug 29 2001

Keywords

Crossrefs

Cf. A027659 (sixth column).

Formula

a(n) = A035343(n+2, 6) = binomial(n+2, 2)*(n^4+24*n^3+221*n^2+954*n+1080)/(6!/2!), n >= 0.
G.f.: (3-2*x-2*x^2+3*x^3-x^4)/(1-x)^7; numerator polynomial is N5(6, x) from the array A063422.
a(n) = 3*C(n+2,2) + 10*C(n+2,3) + 10*C(n+2,4) + 5*C(n+2,5) + C(n+2,6) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A064057 Eighth column of quintinomial coefficients.

Original entry on oeis.org

2, 18, 80, 255, 666, 1520, 3144, 6030, 10890, 18722, 30888, 49205, 76050, 114480, 168368, 242556, 343026, 477090, 653600, 883179, 1178474, 1554432, 2028600, 2621450, 3356730, 4261842, 5368248
Offset: 0

Views

Author

Wolfdieter Lang, Aug 29 2001

Keywords

Crossrefs

Cf. A064056 (seventh column).

Formula

a(n) = A035343(n+2, 7)= binomial(n+3, 3)*(n+14)*(n^3+15*n^2+116*n+120)/(7!/3!).
G.f.: (2+2*x-8*x^2+7*x^3-2*x^4 )/(1-x)^8; numerator polynomial is N5(7, x) from the array A063422.
a(n) = 2*C(n+2,2) + 12*C(n+2,3) + 20*C(n+2,4) + 15*C(n+2,5) + 6*C(n+2,6) + C(n+2,7) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

A064058 Ninth column of quintinomial coefficients.

Original entry on oeis.org

1, 15, 85, 320, 951, 2415, 5475, 11385, 22110, 40612, 71214, 120055, 195650, 309570, 477258, 718998, 1061055, 1537005, 2189275, 3070914, 4247617, 5800025, 7826325, 10445175, 13798980, 18057546, 23422140
Offset: 0

Views

Author

Wolfdieter Lang, Aug 29 2001

Keywords

Links

Crossrefs

Cf. A064057 (eighth column), A000575 (tenth column).

Programs

  • Mathematica
    With[{c=8!/4!},Table[(Binomial[n+4,4](n^4+34n^3+451n^2+2874n+1680))/c, {n,0,30}]] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,15,85,320,951,2415,5475,11385,22110},30] (* Harvey P. Dale, Oct 30 2011 *)

Formula

a(n) = A035343(n+2, 8) = binomial(n+4, 4)*(n^4+34*n^3+451*n^2+2874*n+1680)/(8!/4!).
G.f.: (1+6*x-14*x^2+11*x^3-3*x^4)/(1-x)^9; numerator polynomial is N5(8, x) from the array A063422.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) with a(0)=1, a(1)=15, a(2)=85, a(3)=320, a(4)=951, a(5)=2415, a(6)=5475, a(7)=11385, a(8)=22110. - Harvey P. Dale, Oct 30 2011
a(n) = C(n+2,2) + 12*C(n+2,3) + 31*C(n+2,4) + 35*C(n+2,5) + 21*C(n+2,6) + 7*C(n+2,7) + C(n+2,8) (see comment in A213887). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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