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.

A061318 Column 3 of A061314.

Original entry on oeis.org

0, 4, 40, 304, 1720, 7480, 26404, 79240, 209304, 499140, 1095160, 2242504, 4332640, 7966504, 14036260, 23829040, 39156304, 62512740, 97268904, 147902080, 220270120, 321933304, 462529540, 654208504, 912130600, 1255036900, 1705896504, 2292638040, 3048972304, 4015313320
Offset: 0

Views

Author

Henry Bottomley, Apr 24 2001

Keywords

Links

Crossrefs

Cf. A006007, A061314, A061316, A061319.

Programs

  • Mathematica
    CoefficientList[Series[-4x (1+x+22x^2+22x^3+22x^4+x^5+x^6)/(x-1)^9,{x,0,30}],x] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,4,40,304,1720,7480,26404,79240,209304},30] (* Harvey P. Dale, Jul 04 2023 *)

Formula

a(n) = A061316(n) + A006007(n)^2 = 4*A061319(n).
G.f. -4*x*(1+x+22*x^2+22*x^3+22*x^4+x^5+x^6)/(x-1)^9. - R. J. Mathar, Aug 11 2012

Extensions

a(25)-a(29) from Stefano Spezia, Aug 31 2023

A061320 Column 4 of A061314.

Original entry on oeis.org

0, 5, 140, 6080, 186620, 3504380, 43599605, 392515340, 2738219580, 15571795365, 74962059260, 314303754380, 1173239918240, 3966582090380, 12313551210485, 35488970536640, 95826048090080, 244240228906965, 591327577603980, 1367189227172480, 3032433080571020
Offset: 0

Views

Author

Henry Bottomley, Apr 24 2001

Keywords

Crossrefs

Cf. A061314, A061318, A061319, A061321.

Formula

a(n) = A061318(n) + A061319(n)^2 = 5*A061321(n).
G.f.: -5*x*(1 +11*x +876*x^2 +19780*x^3 +215084*x^4 +1114665*x^5 +2936419*x^6 +4038928*x^7 +2936419*x^8 +1114665*x^9 +215084*x^10 +19780*x^11 +876*x^12 +11*x^13+x^14) / (x-1)^17. - R. J. Mathar, Aug 11 2012
a(n) = n *(n+1) *(n^8+4*n^7+14*n^6+28*n^5+77*n^4+112*n^3+196*n^2+144*n+2304) *(n^2+n+4) * (n^4+2*n^3+5*n^2+4*n+36)/331776. - R. J. Mathar, Aug 11 2012

A061316 a(n) = n*(n+1)*(n^2 + n + 4)/4.

Original entry on oeis.org

0, 3, 15, 48, 120, 255, 483, 840, 1368, 2115, 3135, 4488, 6240, 8463, 11235, 14640, 18768, 23715, 29583, 36480, 44520, 53823, 64515, 76728, 90600, 106275, 123903, 143640, 165648, 190095, 217155, 247008, 279840, 315843, 355215, 398160, 444888
Offset: 0

Views

Author

Henry Bottomley, Apr 24 2001

Keywords

Links

Crossrefs

Cf. A000217, A005563, A006007, A061314.

Programs

  • Maple
    a:=n->sum((n+j^3),j=0..n): seq(a(n),n=0..36); # Zerinvary Lajos, Jul 27 2006
with(combinat):a:=n->sum(fibonacci(4,i), i=0..n): seq(a(n), n=0..36); # Zerinvary Lajos, Mar 20 2008
  • Mathematica
    s=0;lst={};Do[s+=n^3+n*2;AppendTo[lst,s],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 04 2009 *)
Table[n(n+1)(n^2+n+4)/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1}, {0,3,15,48,120},40] (* Harvey P. Dale, May 03 2011 *)
  • PARI
    a(n) = n*(n + 1)*(n^2 + n + 4)/4 \\ Harry J. Smith, Jul 21 2009

Formula

a(n) = n*(n+1)*(n^2 + n + 4)/4.
a(n) = A005563(A000217(n)) = 3*A006007(n) = A061314(n, 2).
a(0)=0, a(1)=3, a(2)=15, a(3)=48, a(4)=120, a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + a(n-5).
G.f.: (-3 (x + x^3))/(-1 + x)^5. - Harvey P. Dale, May 03 2011
Sum_{n>=1} 1/a(n) = 5/4 - tanh(sqrt(15)*Pi/2)*Pi/sqrt(15). - Amiram Eldar, Aug 20 2022
E.g.f.: exp(x)*x*(12 + 18*x + 8*x^2 + x^3)/4. - Stefano Spezia, Aug 31 2023

A061315 Array read by antidiagonals: T(n,k)=T(n,k-1)*(T(n,k-1)+k-1)/k with T(n,1)=n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 5, 6, 4, 1, 10, 16, 10, 5, 1, 28, 76, 40, 15, 6, 1, 154, 1216, 430, 85, 21, 7, 1, 3520, 247456, 37324, 1870, 161, 28, 8, 1, 1551880
Offset: 1

Views

Author

Henry Bottomley, Apr 24 2001

Keywords

Comments

Not always an integer

Examples

			1,1,1,1,1,1,1,1,1,1,...
2,3,5,10,28,154,3520,1551880,267593772160,7160642690122633501504,...
3,6,16,76,1216,247456,61235956672/7,468730299066952899064/49,...
4,10,40,430,37324,232211266,7703153350084336,7417321441864447837991470393906,
5,15,85,1870,700876,81871778626,957569733696568731376,...
6,21,161,6601,8719921,12672844307641,22942997549397847673832961,...
7,28,280,19810,78503068,1027122012987994,...

Crossrefs

Rows include A000012 and A003504. Columns include A000027, A000217, A006007, A061319 and A061321.

Formula

a(n) =A061314(n, k-1)/k

A061322 a(n) = a(n-1) * (1 + a(n-1)/n^2) with a(0) = 2.

Original entry on oeis.org

2, 6, 15, 40, 140, 924, 24640, 12415040, 2408343949440, 71606426901226335015040, 51274803735606705472274088614112357905277056, 21728144612603201307908899563300049012978385050783094682272184269369267136230071558272
Offset: 0

Views

Author

Henry Bottomley, Apr 24 2001

Keywords

Comments

Only first 42 terms are integers (see A003504).

Examples

			a(2) = 6 * (1 + 6/2^2) = 15.

Links

Programs

  • Mathematica
    Block[{n = 0}, NestList[#*(1 + #/++n^2) &, 2, 11]] (* Paolo Xausa, Apr 17 2024 *)
  • PARI
    {a(n) = local(x); if( n<1, 2 * (n==0), (x = a(n-1)) + (x/n)^2)} /* Michael Somos, Apr 02 2006 */

Formula

a(n) = a(n-1) + A003504(n+1)^2, a(n-1) = n * A003504(n+1). a(n) = A061314(2, n).
Showing 1-5 of 5 results.

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