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 1-3 of 3 results.

A154323 Central coefficients of number triangle A113582.

Original entry on oeis.org

1, 2, 10, 37, 101, 226, 442, 785, 1297, 2026, 3026, 4357, 6085, 8282, 11026, 14401, 18497, 23410, 29242, 36101, 44101, 53362, 64010, 76177, 90001, 105626, 123202, 142885, 164837, 189226, 216226, 246017, 278785, 314722, 354026, 396901, 443557, 494210, 549082, 608401, 672401, 741322, 815410, 894917, 980101
Offset: 0

Views

Author

Paul Barry, Jan 07 2009

Keywords

Comments

a(n) equals n!^3 times the determinant of the n X n matrix whose (i,j)-entry is KroneckerDelta[i, j] (((i^3 + 1)/(i^3)) - 1) + 1. - John M. Campbell, May 20 2011
Let b(0)=b(1)=1; b(n)=max(b(n-1)+(n-1)^3, b(n-2)+(n-2)^3); then a(n)=b(n+1). - Yalcin Aktar, Jul 28 2011
a(n-1) is the number of sets of n words of length n over binary alphabet where the first letter occurs n times. a(2) = 10: {aab,abb,bbb}, {aab,bab,bbb}, {aab,bba,bbb}, {aba,abb,bbb}, {aba,bab,bbb}, {aba,bba,bbb}, {abb,baa,bbb}, {abb,bab,bba}, {baa,bab,bbb}, {baa,bba,bbb}. - Alois P. Heinz, Feb 16 2023

Links

Crossrefs

Cf. A000537, A113582, A360660.
Main diagonal of A360693.

Programs

  • Magma
    [(n^4 + 2*n^3 + n^2 + 4)/4: n in [0..40]]; // Vincenzo Librandi, Feb 13 2015
  • Mathematica
    s = 1; lst = {s}; Do[s += n^3; AppendTo[lst, s], {n, 1, 42, 1}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
Table[n!^3*Det[Array[KroneckerDelta[#1,#2](((#1^3+1)/(#1^3))-1)+1&,{n,n}]],{n,1,30}] (* John M. Campbell, May 20 2011 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 2, 10, 37, 101}, 25] (* or *) Table[(n^4 + 2*n^3 + n^2 + 4)/4, {n,0,25}] (* G. C. Greubel, Sep 11 2016 *)

Formula

a(n) = (n^4 + 2*n^3 + n^2 + 4)/4.
G.f.: (1 - 3*x + 10*x^2 - 3*x^3 + x^4)/(1-x)^5.
a(n) = 1 + C(n+1,2)^2 = 1 + A000537(n).
From G. C. Greubel, Sep 11 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
E.g.f.: (1/4)*(4 + 4*x + 14*x^2 + 8*x^3 + x^4)*exp(x). (End)
a(n) = a(n-1)+n^3. - Charles U. Lonappan, Jun 09 2021

A154324 Diagonal sums of number triangle A113582.

Original entry on oeis.org

1, 1, 2, 3, 6, 12, 23, 43, 74, 124, 195, 300, 441, 637, 890, 1226, 1647, 2187, 2848, 3673, 4664, 5874, 7305, 9021, 11024, 13390, 16121, 19306, 22947, 27147, 31908, 37348, 43469, 50405, 58158, 66879, 76570, 87400, 99371, 112671, 127302, 143472, 161183, 180664
Offset: 0

Views

Author

Paul Barry, Jan 07 2009

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{3,0,-8,6,6,-8,0,3,-1}, {1,1,2,3,6,12,23,43,74}, 25] (* G. C. Greubel, Sep 11 2016 *)
CoefficientList[Series[(1 - 2 x - x^2 + 5 x^3 - x^4 - 2 x^5 + x^6) / ((1 - x) (1 - x^2))^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 12 2016 *)
  • PARI
    Vec((1-2*x-x^2+5*x^3-x^4-2*x^5+x^6) / ((1-x)^6*(1+x)^3) + O(x^60)) \\ Colin Barker, Sep 12 2016

Formula

G.f.: (1 -2*x -x^2 +5*x^3 -x^4 -2*x^5 +x^6)/((1-x)*(1-x^2))^3.
a(n) = Sum_{k=0..floor(n/2)} ( 1 + C(k+1,2)*C(n-2k+1,2) ).
From Colin Barker, Sep 12 2016: (Start)
a(n) = (2895 + 945*(-1)^n + (1786-90*(-1)^n)*n - 30*(3+(-1)^n)*n^2 + 40*n^3 + 30*n^4 + 4*n^5)/3840.
a(n) = (2*n^5+15*n^4+20*n^3-60*n^2+848*n+1920)/1920 for n even.
a(n) = (2*n^5+15*n^4+20*n^3-30*n^2+938*n+975)/1920 for n odd. (End)

A154322 a(n) = 1 + n + binomial(n+3,5).

Original entry on oeis.org

1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298, 2014, 3016, 4382, 6203, 8584, 11645, 15522, 20368, 26354, 33670, 42526, 53153, 65804, 80755, 98306, 118782, 142534, 169940, 201406, 237367, 278288, 324665, 377026, 435932, 501978, 575794, 658046, 749437, 850708
Offset: 0

Views

Author

Paul Barry, Jan 07 2009

Keywords

Comments

Row sums of number triangle A113582.
It appears that the sequence is the pairwise sum of terms in A101338 and A000389 with offsets as follows:
1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298, ... =
1, 2, 4, 9, 20, 41, 77, 134, 219, 340, 506, ... +
0, 0, 0, 1, 6, 21, 56, 126, 252, 462, 792, ...
- Gary W. Adamson, Oct 08 2015

Links

Crossrefs

Cf. A000389, A101338, A113582.

Programs

  • Magma
    [1+n+Binomial(n+3,5) : n in [0..50]]; // Wesley Ivan Hurt, Oct 08 2015
  • Magma
    I:=[1,2,4,10,26,62]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Oct 09 2015
  • Maple
    A154322:=n->1+n+binomial(n+3,5): seq(A154322(n), n=0..50); # Wesley Ivan Hurt, Oct 08 2015
  • Mathematica
    CoefficientList[Series[(1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1 - x)^6, {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 08 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 2, 4, 10, 26, 62}, 50] (* Vincenzo Librandi, Oct 09 2015 *)
 Table[ (n + 1)*(n^4 + 4*n^3 + n^2 - 6*n + 120)/120 , {n, 0, 25}] (* G. C. Greubel, Sep 10 2016 *)
Table[1+n+Binomial[n+3,5],{n,0,40}] (* Harvey P. Dale, Jan 19 2023 *)
  • PARI
    Vec((1-4*x+7*x^2-4*x^3+x^4)/(1-x)^6 + O(x^100)) \\ Altug Alkan, Oct 18 2015

Formula

G.f.: (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1-x)^6;
a(n) = n + 1 + Sum_{k=0..n} binomial(k+1,2) * binomial(n-k+1,2).
a(n) = (n+1)*(n^4 +4*n^3 +n^2 -6*n +120)/120.
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) for n>5. - Wesley Ivan Hurt, Oct 08 2015
E.g.f.: (1/120)*(120 + 120*x + 60*x^2 + 60*x^3 + 15*x^4 + x^5)*exp(x). - G. C. Greubel, Sep 10 2016
Showing 1-3 of 3 results.

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