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.

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A295181 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x)^k.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 3, 4, 9, 0, 1, 0, 4, 6, 24, 44, 0, 1, 0, 5, 8, 45, 128, 265, 0, 1, 0, 6, 10, 72, 252, 880, 1854, 0, 1, 0, 7, 12, 105, 416, 1935, 6816, 14833, 0, 1, 0, 8, 14, 144, 620, 3520, 16146, 60032, 133496, 0, 1, 0, 9, 16, 189, 864, 5725, 31104, 153657, 589312, 1334961, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 16 2017

Keywords

Comments

A(n,k) is the k-fold exponential convolution of A000166 with themselves, evaluated at n.

Examples

			E.g.f. of column k: A_k(x) = 1 + k*x^2/2! + 2*k*x^3/3! + 3*k*(k + 2)*x^4/4! + 4*k*(5*k + 6)*x^5/5! + 5*k*(3*k^2 + 26*k + 24)*x^6/6! + ...
Square array begins:
  1,   1,    1,    1,    1,    1,  ...
  0,   0,    0,    0,    0,    0,  ...
  0,   1,    2,    3,    4,    5,  ...
  0,   2,    4,    6,    8,   10,  ...
  0,   9,   24,   45,   72,  105,  ...
  0,  44,  128,  252,  416,  620,  ...

Links

Crossrefs

Columns k=0..5 give A000007, A000166, A087981, A137775, A383344, A383384.
Rows n=0..3 give A000012, A000004, A001477, A005843.
Main diagonal gives A295182.
Cf. A008279, A265609.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
  • PARI
    a(n, k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k-1, j)/(n-j)!); \\ Seiichi Manyama, Apr 25 2025

Formula

E.g.f. of column k: exp(-k*x)/(1 - x)^k.
From Seiichi Manyama, Apr 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k-1,j)/(n-j)!.
A(0,k) = 1, A(1,k) = 0; A(n,k) = (n-1) * (A(n-1,k) + k*A(n-2,k)). (End)

A356546 Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).

Original entry on oeis.org

1, 2, 2, 6, 12, 6, 20, 60, 60, 20, 70, 280, 420, 280, 70, 252, 1260, 2520, 2520, 1260, 252, 924, 5544, 13860, 18480, 13860, 5544, 924, 3432, 24024, 72072, 120120, 120120, 72072, 24024, 3432, 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870
Offset: 0

Views

Author

Peter Luschny, Aug 12 2022

Keywords

Comments

The counterpart using the falling factorial is Leibniz's Harmonic Triangle A003506.

Examples

			Triangle T(n, k) begins:
[0]     1;
[1]     2,      2;
[2]     6,     12,      6;
[3]    20,     60,     60,     20;
[4]    70,    280,    420,    280,     70;
[5]   252,   1260,   2520,   2520,   1260,    252;
[6]   924,   5544,  13860,  18480,  13860,   5544,    924;
[7]  3432,  24024,  72072, 120120, 120120,  72072,  24024,   3432;
[8] 12870, 102960, 360360, 720720, 900900, 720720, 360360, 102960, 12870;

Crossrefs

cf. A000984, A059304 (row sums, see also A343842), A265609 (rising factorial).
Cf. A003506, A173018 (Eulerian numbers), A000108, A000897 (central terms).

Programs

  • Maple
    A356546 := (n, k) -> pochhammer(n+1, n)/(k!*(n-k)!):
for n from 0 to 8 do seq(A356546(n, k), k=0..n) od;
  • Mathematica
    T[ n_, k_] := Binomial[2*n, n] * Binomial[n, k]; (* Michael Somos, Aug 18 2022 *)
  • PARI
    {T(n, k) = binomial(2*n, n) * binomial(n, k)}; /* Michael Somos, Aug 18 2022 */
  • SageMath
    def A356546(n, k):
    return rising_factorial(n+1,n) // (factorial(k) * factorial(n-k))
for n in range(9): print([A356546(n, k) for k in range(n+1)])

Formula

Bernoulli(n) / Catalan(n) = Sum_{k=0..n} (-1)^k*A173018(n, k) / T(n, k), (with Bernoulli(1) = 1/2).
G.f.: 1/sqrt(1 - 4*x*(y + 1)). - Vladimir Kruchinin, Feb 15 2023

A326326 T(n, k) = [x^k] Sum_{j=0..n} Pochhammer(x, j), for 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 15, 7, 1, 1, 34, 65, 42, 11, 1, 1, 154, 339, 267, 96, 16, 1, 1, 874, 2103, 1891, 831, 191, 22, 1, 1, 5914, 15171, 15023, 7600, 2151, 344, 29, 1, 1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1
Offset: 0

Views

Author

Peter Luschny, Jul 02 2019

Keywords

Examples

			Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 2,     1]
[3] [1, 4,     4,      1]
[4] [1, 10,    15,     7,      1]
[5] [1, 34,    65,     42,     11,    1]
[6] [1, 154,   339,    267,    96,    16,    1]
[7] [1, 874,   2103,   1891,   831,   191,   22,   1]
[8] [1, 5914,  15171,  15023,  7600,  2151,  344,  29,  1]
[9] [1, 46234, 124755, 133147, 74884, 24600, 4880, 575, 37, 1]

Crossrefs

Same construction for the falling factorial is A176663.
The inverse of the lower triangular matrix is the signed form of A256894.
Second column is A003422(n) and row sums are A003422(n+1).
Alternating row sums are A000007.
Third column is A097422.
Cf. A265609, A132393.

Programs

  • Maple
    with(PolynomialTools):
T_row := n -> CoefficientList(expand(add(pochhammer(x, j), j=0..n)),x):
ListTools:-Flatten([seq(T_row(n), n=0..9)]);
  • Mathematica
    Table[CoefficientList[FunctionExpand[Sum[Pochhammer[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten

Formula

Sum_{k=0..n} T(n, k)*x^k = Sum_{k=0..n} (x)^k, where (x)^k denotes the rising factorial.
Conjecture: T(n,k) = Sum_{i=0..n} A132393(i,k) for 0 <= k <= n. - Werner Schulte, Mar 30 2022

A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

Original entry on oeis.org

2, 3, 8, 12, 30, 60, 144, 330, 120, 840, 2100, 1260, 5760, 15344, 11760, 1680, 45360, 127008, 113400, 30240, 403200, 1176120, 1169280, 428400, 30240, 3991680, 12054240, 13000680, 5821200, 831600, 43545600, 135508032, 155923680, 80415720, 16632000, 665280
Offset: 2

Views

Author

Steven Finch, Nov 13 2021

Keywords

Comments

A round means the same as a directed ring or circle.

Examples

			Triangle starts:
[2]     2;
[3]     3;
[4]     8,     12;
[5]    30,     60;
[6]   144,    330,    120;
[7]   840,   2100,   1260;
[8]  5760,  15344,  11760,  1680;
[9] 45360, 127008, 113400, 30240;
...
For n = 4, there are 8 ways to make one round and 12 ways to make two rounds.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Links

Crossrefs

Row sums give A066166 (Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A001813(n/2) = |A067994(n)| for even n.
Cf. A008279, A265609, A331327.

Programs

  • Maple
    ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):
T := (n, k) -> n!*coeff(xcoeff(n), t, k):
seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # Peter Luschny, Nov 13 2021
# second Maple program:
A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):
seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # Mélika Tebni, May 03 2023
  • Mathematica
    f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]

Formula

G.f.: (1 - x)^(-x*t).
T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023
The above formula can also be written as T(n, k) = A008279(n, k)*A331327(n, k) or as T(n, k) = A265609(n + 1, k)*A331327(n, k). - Peter Luschny, May 03 2023

A345406 Integers k such that k = d1^(c) + d2^(c) + ... + dc^(c), where d^(c) denotes the rising factorial of d, c is the length of k and di is the i-th digit of k in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 90, 744, 840
Offset: 1

Views

Author

Andrzej Kukla, Jun 18 2021

Keywords

Comments

The rising factorial d^(c) is defined as d*(d+1)*(d+2)*...*(d+c-1).

Examples

			7^(3) + 4^(3) + 4^(3) = 7*8*9 + 4*5*6 + 4*5*6 = 504 + 120 + 120 = 744, therefore 744 is in the list.

Crossrefs

Cf. A014080 (factorions), A265609 (rising factorials), A345405.

Programs

  • Mathematica
    q[n_] := Module[{dig = IntegerDigits[n], nd}, nd = Length[dig]; Sum[(d + nd - 1)!/(d - 1)!, {d, dig}] == n]; Select[Range[0, 1000], q] (* Amiram Eldar, Jun 18 2021 *)
Previous Showing 11-15 of 15 results.

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