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.

A302353 a(n) = Sum_{k=0..n} k^n*binomial(2*n-k,n).

Original entry on oeis.org

1, 1, 7, 69, 936, 16290, 345857, 8666413, 250355800, 8191830942, 299452606190, 12095028921250, 534924268768540, 25710497506696860, 1334410348734174285, 74379234152676275325, 4431350132232658244400, 281020603194039519937590, 18900157831016574533520330, 1343698678390575915132318870
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 06 2018

Keywords

Comments

a(n) is the n-th term of the main diagonal of iterated partial sums array of n-th powers (starting with the first partial sums).

Examples

			For n = 4 we have:
------------------------
0   1    2    3    [4]
------------------------
0,  1,  17,   98,  354,  ... A000538 (partial sums of fourth powers)
0,  1,  18,  116,  470,  ... A101089 (partial sums of A000538)
0,  1,  19,  135,  605,  ... A101090 (partial sums of A101089)
0,  1,  20,  155,  760,  ... A101091 (partial sums of A101090)
0,  1,  21,  176, [936], ... A254681 (partial sums of A101091)
------------------------
therefore a(4) = 936.

Crossrefs

Cf. A002054, A031971, A265612, A293550, A293574, A302352.

Programs

  • Mathematica
    Join[{1}, Table[Sum[k^n Binomial[2 n - k, n], {k, 0, n}], {n, 19}]]
Table[SeriesCoefficient[HurwitzLerchPhi[x, -n, 0]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 19}]

Formula

a(n) ~ c * (r * (2-r)^(2-r) / (1-r)^(1-r))^n * n^n, where r = 0.69176629470097668698335106516328398961170464277337300459988208658267146... is the root of the equation (2-r) = (1-r) * exp(1/r) and c = 0.96374921279011282619632879505754646526289414675402231447188230355850496... - Vaclav Kotesovec, Apr 08 2018

A371836 a(n) = Sum_{k=0..floor(n/2)} n^k * binomial(2*n-2*k-1,n-2*k).

Original entry on oeis.org

1, 1, 5, 19, 91, 426, 2190, 11467, 63811, 365806, 2200978, 13677962, 88553726, 591576220, 4093814812, 29164567635, 214244414371, 1616044475734, 12523774634922, 99418836782602, 808492937082410, 6720935024074092, 57100849909374340, 495022008799053006
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2024

Keywords

Crossrefs

Cf. A293574, A371837.
Cf. A371798, A371826.

Programs

  • Mathematica
    Join[{1}, Table[Sum[n^k*Binomial[2*n-2*k-1,n-1], {k, 0, n/2}], {n, 1, 25}]] (* Vaclav Kotesovec, Apr 08 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, n^k*binomial(2*n-2*k-1, n-2*k));

Formula

a(n) = [x^n] 1/((1-n*x^2) * (1-x)^n).
a(n) ~ exp(sqrt(n) + 1/2) * n^(n/2) / 2. - Vaclav Kotesovec, Apr 08 2024

A371837 a(n) = Sum_{k=0..floor(n/3)} n^k * binomial(2*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 1, 3, 13, 51, 201, 834, 3529, 15075, 65431, 288278, 1285263, 5799470, 26492103, 122432628, 572291385, 2705760291, 12937116213, 62542367166, 305668511259, 1510080076410, 7539381024297, 38034307340076, 193835252945487, 997724306958606, 5185731234177001
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2024

Keywords

Crossrefs

Cf. A293574, A371836.
Cf. A120305, A371827.

Programs

  • Mathematica
    Join[{1}, Table[Sum[n^k*Binomial[2*n-3*k-1,n-1], {k, 0, n/3}], {n, 1, 25}]] (* Vaclav Kotesovec, Apr 08 2024 *)
  • PARI
    a(n) = sum(k=0, n\3, n^k*binomial(2*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1-n*x^3) * (1-x)^n).
a(n) ~ exp(n^(2/3) + n^(1/3)/2 + 1/3) * n^(n/3) / 3. - Vaclav Kotesovec, Apr 08 2024

A368506 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(j+k-1,j).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 11, 4, 0, 1, 8, 24, 26, 5, 0, 1, 10, 42, 82, 57, 6, 0, 1, 12, 65, 188, 261, 120, 7, 0, 1, 14, 93, 360, 787, 804, 247, 8, 0, 1, 16, 126, 614, 1870, 3204, 2440, 502, 9, 0, 1, 18, 164, 966, 3810, 9476, 12900, 7356, 1013, 10, 0
Offset: 0

Views

Author

Seiichi Manyama, Dec 27 2023

Keywords

Examples

			Square array begins:
  1, 1,   1,    1,     1,     1,      1, ...
  0, 2,   4,    6,     8,    10,     12, ...
  0, 3,  11,   24,    42,    65,     93, ...
  0, 4,  26,   82,   188,   360,    614, ...
  0, 5,  57,  261,   787,  1870,   3810, ...
  0, 6, 120,  804,  3204,  9476,  23112, ...
  0, 7, 247, 2440, 12900, 47590, 139134, ...

Crossrefs

Columns k=0..3 give A000007, A000027(n+1), A125128(n+1), A052150.
Main diagonal gives A293574.
Cf. A008949, A368487.

Programs

  • PARI
    T(n, k) = sum(j=0, n, k^(n-j)*binomial(j+k-1, j));

Formula

G.f. of column k: 1/((1-k*x) * (1-x)^k).

A368510 a(n) = Sum_{k=0..n} n^(n-k) * binomial(k+n,k).

Original entry on oeis.org

1, 3, 16, 113, 1026, 11782, 166776, 2825349, 55797790, 1258065866, 31866312336, 895430095738, 27632885411236, 928823226029532, 33772464199743184, 1320627875038128045, 55259636489069057910, 2463499964955575965954, 116560977980742613228704
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Crossrefs

Cf. A293574, A368489.

Programs

  • PARI
    a(n) = sum(k=0, n, n^(n-k)*binomial(n+k, k));

Formula

a(n) = [x^n] 1/((1-n*x) * (1-x)^(n+1)).
a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Dec 28 2023
Showing 1-5 of 5 results.

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