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 1-8 of 8 results.

A126885 T(n,k) = n*T(n,k-1) + k, with T(n,1) = 1, square array read by ascending antidiagonals (n >= 0, k >= 1).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 11, 10, 5, 1, 6, 18, 26, 15, 6, 1, 7, 27, 58, 57, 21, 7, 1, 8, 38, 112, 179, 120, 28, 8, 1, 9, 51, 194, 453, 543, 247, 36, 9, 1, 10, 66, 310, 975, 1818, 1636, 502, 45, 10, 1, 11, 83, 466, 1865, 4881, 7279, 4916, 1013, 55, 11
Offset: 0

Views

Author

Gary W. Adamson, Dec 30 2006

Keywords

Examples

			Square array begins:
  n\k | 1   2   3   4    5     6      7       8 ...
  -------------------------------------------------
    0 | 1   2   3   4    5     6      7       8 ... A000027
    1 | 1   3   6  10   15    21     28      36 ... A000217
    2 | 1   4  11  26   57   120    247     502 ... A000295
    3 | 1   5  18  58  179   543   1636    4916 ... A000340
    4 | 1   6  27 112  453  1818   7279   29124 ... A014825
    5 | 1   7  38 194  975  4881  24412  122068 ... A014827
    6 | 1   8  51 310 1865 11196  67183  403106 ... A014829
    7 | 1   9  66 466 3267 22875 160132 1120932 ... A014830
    8 | 1  10  83 668 5349 42798 342391 2739136 ... A014831
    ...

Crossrefs

Antidiagonal sums are A134195.
Main diagonal gives A062805.
Cf. A000027, A000217, A000295, A000340, A014825, A014827, A014829, A014830, A014831.

Programs

  • Maxima
    T(n, k) := if k = 1 then 1 else n*T(n, k - 1) + k$
create_list(T(n - k + 1, k), n, 0, 20, k, 1, n + 1);
/* Franck Maminirina Ramaharo, Jan 26 2019 */

Formula

T(1,k) = k*(k + 1)/2, and T(n,k) = (k - (k + 1)*n + n^(k + 1))/(n^2 - 2*n + 1) elsewhere.
T(n,k) = third entry in the vector M^k * (1, 0, 0), where M is the following 3 X 3 matrix:
1, 0, 0
1, 1, 0
1, 1, n.

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Jan 26 2019

A068475 a(n) = Sum_{m=0..n} m*n^(m-1).

Original entry on oeis.org

0, 1, 5, 34, 313, 3711, 54121, 937924, 18831569, 429794605, 10987654321, 310989720966, 9652968253897, 326011399456939, 11901025061692313, 466937872906120456, 19594541482740368161, 875711370981239308953, 41524755927216069067489, 2082225625247428808306410
Offset: 0

Views

Author

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

Keywords

Comments

The closed form comes from taking the derivative of the closed form of A031972, for which each term of this sequence is a derivative. - Jonas Whidden, Oct 18 2011

Examples

			a(2) = Sum_{m = 1..2} m*2^(m-1) = 1 + 2*2 = 5.

Links

Crossrefs

Derivative sequence of A031972.
Cf. A023037, A062805, A062806, A368534.

Programs

  • Haskell
    a068475 n = sum $ zipWith (*) [1..n] $ iterate (* n) 1
-- Reinhard Zumkeller, Nov 22 2014
  • Magma
    [0] cat [(&+[m*n^(m-1): m in [0..n]]): n in [1..30]]; // G. C. Greubel, Oct 13 2018
  • Maple
    a := n->sum(m*n^(m-1),m=1..n);
  • Mathematica
    Join[{0}, Table[Sum[m*n^(m-1), {m,0,n}], {n,1,30}]] (* G. C. Greubel, Oct 13 2018 *)
  • PARI
    for(n=0,30, print1(if(n==0, 0, sum(m=0,n, m*n^(m-1))), ", ")) \\ G. C. Greubel, Oct 13 2018

Formula

a(1) = 1. For n > 1, a(n) = ((n-1)*(n+1)*n^n - n^(n+1) + 1)/(n-1)^2. - Jonas Whidden, Oct 18 2011
a(n) = A062806(n) / n for n>=1. - Reinhard Zumkeller, Nov 22 2014
a(n) = [x^(n-1)] 1/((1 - x)*(1 - n*x)^2). - Peter Bala, Feb 12 2024

A368524 a(n) = Sum_{k=1..n} k^2 * n^(n-k).

Original entry on oeis.org

0, 1, 6, 30, 180, 1455, 15666, 213500, 3521736, 68101245, 1508916310, 37661140506, 1045012524348, 31900040161899, 1062139933257690, 38299757176168440, 1486670929792295696, 61800664096000744569, 2738952078516469743678, 128909373997071187219990
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Crossrefs

Cf. A062805, A368505, A368525.
Cf. A368526, A368534.

Programs

  • PARI
    a(n) = sum(k=1, n, k^2*n^(n-k));

Formula

a(n) = [x^n] x * (1+x)/((1-n*x) * (1-x)^3).
a(n) = n * (n+1) * (n^n - n^2 + n - 1)/(n-1)^3 for n > 1.

A368525 a(n) = Sum_{k=1..n} k^3 * n^(n-k).

Original entry on oeis.org

0, 1, 10, 60, 364, 2745, 27246, 346864, 5422264, 100449225, 2149062490, 52097910876, 1410401518692, 42153624499441, 1378058477508454, 48900582823143360, 1871456346915007216, 76821658841556480753, 3366451935514051046802, 156839738363103277783900
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Crossrefs

Cf. A062805, A368505, A368524.
Cf. A368527.

Programs

  • PARI
    a(n) = sum(k=1, n, k^3*n^(n-k));

Formula

a(n) = [x^n] x * (1+4*x+x^2)/((1-n*x) * (1-x)^4).
a(n) = n * (n^n * (n^2 + 4*n + 1) - n^5 - 3*n^2 - n - 1)/(n-1)^4 for n > 1.

A368534 a(n) = Sum_{k=1..n} binomial(k+1,2) * n^(n-k).

Original entry on oeis.org

0, 1, 5, 24, 146, 1215, 13431, 186816, 3130436, 61291125, 1371742105, 34522712136, 964626945558, 29621465864627, 991330604373851, 35906022352657920, 1399219698628043016, 58367293868445147657, 2594796705962971336125, 122463905297217627859000
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2023

Keywords

Crossrefs

Cf. A062805, A368535.
Cf. A002662, A052150, A052161.
Cf. A023037, A062806, A068475.

Programs

  • Mathematica
    Table[Sum[Binomial[k+1,2]n^(n-k),{k,n}],{n,0,20}] (* Harvey P. Dale, May 14 2025 *)
  • PARI
    a(n) = sum(k=1, n, binomial(k+1, 2)*n^(n-k));

Formula

a(n) = [x^n] x/((1-n*x) * (1-x)^3).
a(n) = n * (2*n^(n+1) - n^3 - n^2 + n - 1)/(2 * (n-1)^3) for n > 1.

A089000 Square table, read by antidiagonals, of coefficients T(k,n) (row k; column n) defined by: T(k,n) = k*T(k,n-1)+ n; T(k,0) = 0.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 4, 1, 0, 5, 10, 11, 5, 1, 0, 6, 15, 26, 18, 6, 1, 0, 7, 21, 57, 58, 27, 7, 1, 0, 8, 28, 120, 179, 112, 38, 8, 1, 0, 9, 36, 247, 543, 453, 194, 51, 9, 1, 0, 10, 45, 502, 1636, 1818, 975, 310, 66, 10, 1, 0
Offset: 0

Views

Author

Philippe Deléham, Nov 02 2003

Keywords

Crossrefs

Rows begin:
{0, 1, 2, 3, 4, 5, 6, 7, 8, ...}:see A001477
{0, 1, 3, 6, 10, 15, 21, 28, ...} : see A000217
{0, 1, 4, 11, 26, 57, 120, 247, 502, ...} : see A000295
{0, 1, 5, 18, 58, 179, 543, 1636, ...} : see A000340
{0, 1, 6, 27, 112, 453, 1818, 7279, ...} : see A014825
{0, 1, 7, 38, 194, 975, 4881, 24412, ...} : see A014827
{0, 1, 8, 51, 310, 1865, 11196, 67183, ...}: see diagonals of triangle A088990
Diagonal begin:
{0, 1, 4, 18, 112, 975, 11196, ... } :see A062805
{0, 1, 5, 27, 194, 1865, ...} : see A023811
Column {3, 6, 11, 18, 27, 38, 51, ...} : see A010000

Programs

  • Mathematica
    Unprotect[Power]; 0^0=1; T[n_,k_]:=Sum[j*k^(n-j),{j,0,n}]; Table[T[n-k,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Apr 19 2025 *)

Formula

T(k, n)= (k^(n+1)- (k-1)*n - k)/(k-1)^2. T(k, n) = Sum(j, 0<=j<=n; j*k^(n-j)).

A347274 a(n) = Sum_{j=1..n} j*n^(n+1-j).

Original entry on oeis.org

1, 8, 54, 448, 4875, 67176, 1120924, 21913088, 490329045, 12345679000, 345227121426, 10610896401216, 355457590375615, 12887297856860168, 502684312937211000, 20988295479420645376, 933876701895122362665, 44111544001370512714296, 2204350295349917301462190
Offset: 1

Views

Author

Ryan Stubbs, Aug 25 2021

Keywords

Examples

			a(1) = 1;
a(2) = 2^2 + 2*2^1 = 8;
a(3) = 3^3 + 2*3^2 + 3*3^1 = 54;
a(4) = 4^4 + 2*4^3 + 3*4^2 + 4*4^1 = 448;
a(5) = 5^5 + 2*5^4 + 3*5^3 + 4*5^2 + 5*5^1 = 4875.

Crossrefs

Cf. A058128, A060073, A062805.

Programs

  • Maple
    a:= n-> `if`(n=1, 1, (n^n-n)*(n/(n-1))^2):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 02 2021
  • Python
    def A347274(n): return 1 if n == 1 else n**2*(n**n-n)//(n - 1)**2 # Chai Wah Wu, Sep 12 2021

Formula

a(n) = (n^n - n)*(n/(n-1))^2 for n > 1, a(1) = 1.
a(n) = n * A062805(n) = n^2 * A058128(n) = n^3 * A060073(n).

A368535 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^(n-k).

Original entry on oeis.org

0, 1, 6, 31, 188, 1510, 16106, 217938, 3577624, 68952495, 1524157870, 37974983321, 1052320304212, 32089921353308, 1067586804710258, 38470738234990580, 1492501011869912496, 62015249735222969325, 2747431806313734355830, 129267455591507496073315
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2023

Keywords

Crossrefs

Cf. A062805, A368534.
Cf. A002663, A097786, A097788.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(k+2, 3)*n^(n-k));

Formula

a(n) = [x^n] x/((1-n*x) * (1-x)^4).
a(n) = n * (6*n^(n+2) - n^5 - 3*n^4 + n^3 + n^2 - 6*n + 2)/(6 * (n-1)^4) for n > 1.
Showing 1-8 of 8 results.

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