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 1-7 of 7 results.

A092987 A092985(n) divided by {A057237(n)}!.

Original entry on oeis.org

1, 3, 14, 585, 924, 1339975, 164604, 14454403425, 1163340147200, 478015854767451, 33644021190, 37947924636264267625, 29447897812956, 6178966019176767549393375, 1575167029892595726671372288
Offset: 1

Views

Author

Amarnath Murthy, Mar 28 2004

Keywords

Examples

			a(7) = 164604, as A092985(7) = 118514880 and A057237(n) = 6
118514880/720 =164604.

Crossrefs

Cf. A092985, A057237.

Extensions

More terms from David Wasserman, Aug 22 2006

A256268 Table of k-fold factorials, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 15, 4, 1, 1, 1, 120, 105, 28, 5, 1, 1, 1, 720, 945, 280, 45, 6, 1, 1, 1, 5040, 10395, 3640, 585, 66, 7, 1, 1, 1, 40320, 135135, 58240, 9945, 1056, 91, 8, 1, 1, 1, 362880, 2027025, 1106560, 208845, 22176, 1729, 120, 9, 1, 1
Offset: 0

Views

Author

Philippe Deléham, Jun 01 2015

Keywords

Comments

A variant of A142589.

Examples

			1  1   1    1     1       1         1... A000012
1  1   2    6    24     120       720... A000142
1  1   3   15   105     945     10395... A001147
1  1   4   28   280    3640     58240... A007559
1  1   5   45   585    9945    208845... A007696
1  1   6   66  1056   22176    576576... A008548
1  1   7   91  1729   43225   1339975... A008542
1  1   8  120  2640   76560   2756160... A045754
1  1   9  153  3825  126225   5175225... A045755
1  1  10  190  5320  196840   9054640... A045756
1  1  11  231  7161  293601  14977651... A144773

Links

Crossrefs

Cf. Columns : A000012, A000012, A000384, A011199, A011245.
Cf. Diagonals : A092985, A076111, A158887.
Cf. A048994, A132393.
Cf. A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A144773 (10)

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Product([0..n-k-1], j-> j*k+1) ))); # G. C. Greubel, Mar 04 2020
  • Magma
    function T(n,k)
  if k eq 0 or n eq 0 then return 1;
  else return (&*[j*k+1: j in [0..n-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 04 2020
  • Maple
    seq(seq( mul(j*k+1, j=0..n-k-1), k=0..n), n=0..12); # G. C. Greubel, Mar 04 2020
  • Mathematica
    T[n_, k_]= Product[j*k+1, {j,0,n-1}]; Table[T[n-k,k], {n,0,12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 04 2020 *)
  • PARI
    T(n,k) = prod(j=0, n-1, j*k+1);
for(n=0,12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Mar 04 2020
  • Sage
    [[ product(j*k+1 for j in (0..n-k-1)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 04 2020

Formula

A(n, k) = (-n)^k*FallingFactorial(-1/n, k) for n >= 1. - Peter Luschny, Dec 21 2021

A076111 Product of terms in n-th row of A076110.

Original entry on oeis.org

1, 2, 15, 280, 9945, 576576, 49579075, 5925744000, 939536222625, 190787784140800, 48279601331512551, 14894665739501184000, 5502449072258318805625, 2397950328737212204032000
Offset: 0

Views

Author

Amarnath Murthy, Oct 09 2002

Keywords

Links

Crossrefs

Cf. A002522, A064808, A076110.

Programs

  • GAP
    List([0..15], n-> Product([1..n], j-> j*n+1) ); # G. C. Greubel, Mar 04 2020
  • Magma
    [1] cat [&*[j*n+1: j in [1..n]]: n in [1..15]]; // G. C. Greubel, Mar 04 2020
  • Maple
    seq( mul(j*n+1, j=1..n), n=0..15); # G. C. Greubel, Mar 04 2020
  • Mathematica
    Table[Product[j*n+1, {j,n}], {n,0,15}] (* G. C. Greubel, Mar 04 2020 *)
  • Maxima
    A076111(n):=prod(1+n*k,k,1,n)$
makelist(A076111(n),n,0,30); /* Martin Ettl, Nov 07 2012 */
  • PARI
    vector(16, n, my(m=n-1); prod(j=1,m, j*m+1)) \\ G. C. Greubel, Mar 04 2020
  • Sage
    [product(j*n+1 for j in (1..n)) for n in (0..15)] # G. C. Greubel, Mar 04 2020

Formula

a(n) = Prod_{k=1..n} (1+n*k). - Yalcin Aktar, Jul 14 2009
a(n) = n^n * Pochhammer(n, 1 + 1/n). - G. C. Greubel, Mar 04 2020
a(n) = A092985(n)*(n^2+1). - R. J. Mathar, Mar 30 2023

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

A349731 a(n) = -(-n)^n * FallingFactorial(1/n, n) for n >= 1 and a(0) = -1.

Original entry on oeis.org

-1, 1, 1, 10, 231, 9576, 623645, 58715280, 7547514975, 1270453824640, 271252029133449, 71635824470246400, 22929813173612997575, 8747686347650933760000, 3921812703436118765113125, 2041590849971133677650610176, 1221367737152989777782325269375, 832163138229382457228044554240000
Offset: 0

Views

Author

Peter Luschny, Dec 21 2021

Keywords

Links

Crossrefs

The main diagonal of A349971 for n >= 1.
The Stirling set counterpart is A318183.
Cf. A008279, A092985, A132393.

Programs

  • Magma
    [-1,1] cat [Round(n^(n-1)*Gamma((n^2-1)/n)/Gamma((n-1)/n)): n in [2..30]]; // G. C. Greubel, Feb 22 2022
  • Maple
    A349731 := n -> -add((-1)^(n-k)*Stirling1(n, n-k)*(-n)^k, k = 0..n):
seq(A349731(n), n = 0..17);
  • Mathematica
    a[0] = -1; a[n_] := -(-n)^n * FactorialPower[1/n, n]; Array[a, 18, 0] (* Amiram Eldar, Dec 21 2021 *)
  • Python
    from sympy import ff
from fractions import Fraction
def A349731(n): return -1 if n == 0 else -(-n)**n*ff(Fraction(1,n),n) # Chai Wah Wu, Dec 21 2021
  • SageMath
    def a(n): return -(-n)^n*falling_factorial(1/n, n) if n > 0 else -1
print([a(n) for n in (1..17)])

Formula

a(n) = -(-1)^n*Sum_{k=0..n}[n, n-k]*(-n)^k, where [n, k] denotes the Stirling cycle numbers A132393(n, k).

A352074 a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).

Original entry on oeis.org

1, 1, 4, 42, 904, 34070, 2019888, 174588120, 20804747136, 3276218158560, 659664288364800, 165425062846302336, 50574549124825998336, 18520126461205806360144, 8003819275469728355033088, 4031020344281171589447408000, 2340375822778055527109749211136
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

Links

Crossrefs

Cf. A007840, A092985, A094420, A227917, A317171, A317172, A331690, A352069, A352071.
Cf. A081048.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ Michel Marcus, Mar 02 2022

Formula

a(n) = n! * [x^n] 1 / (1 + log(1 - n*x) / n) for n > 0.
a(n) ~ n! * n^(n-2) * (1 + 2*log(n)/n). - Vaclav Kotesovec, Mar 03 2022

A355006 Triangle read by rows. T(n, k) = n^(n - k) * |Stirling1(n, k)|.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 18, 9, 1, 0, 384, 176, 24, 1, 0, 15000, 6250, 875, 50, 1, 0, 933120, 355104, 48600, 3060, 90, 1, 0, 84707280, 29647548, 3899224, 252105, 8575, 147, 1, 0, 10569646080, 3425697792, 430309376, 27725824, 1003520, 20608, 224, 1
Offset: 0

Views

Author

Peter Luschny, Jun 17 2022

Keywords

Examples

			Table T(n, k) begins:
[0] 1;
[1] 0,           1;
[2] 0,           2,          1;
[3] 0,          18,          9,         1;
[4] 0,         384,        176,        24,        1;
[5] 0,       15000,       6250,       875,       50,       1;
[6] 0,      933120,     355104,     48600,     3060,      90,     1;
[7] 0,    84707280,   29647548,   3899224,   252105,    8575,   147,   1;
[8] 0, 10569646080, 3425697792, 430309376, 27725824, 1003520, 20608, 224, 1;

Crossrefs

A152684 (column 1), A006002 (subdiagonal), A092985 (row sums), A355007.

Programs

  • Maple
    seq(seq(n^(n - k)*abs(Stirling1(n, k)), k = 0..n), n = 0..9);
  • Mathematica
    T[n_, k_] := If[n == k == 0, 1, n^(n - k) * Abs[StirlingS1[n, k]]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 17 2022 *)

A368119 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} (n*j + 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 15, 24, 1, 1, 1, 5, 28, 105, 120, 1, 1, 1, 6, 45, 280, 945, 720, 1, 1, 1, 7, 66, 585, 3640, 10395, 5040, 1, 1, 1, 8, 91, 1056, 9945, 58240, 135135, 40320, 1, 1, 1, 9, 120, 1729, 22176, 208845, 1106560, 2027025, 362880, 1
Offset: 0

Views

Author

Peter Luschny, Dec 18 2023

Keywords

Comments

A(n, k) is the number of increasing (n + 1)-ary trees on k vertices. (Following a comment of David Callan in A007559.)

Examples

			Array A(n, k) starts:
  [0] 1, 1, 1,   1,    1,      1,       1,         1, ...  A000012
  [1] 1, 1, 2,   6,   24,    120,     720,      5040, ...  A000142
  [2] 1, 1, 3,  15,  105,    945,   10395,    135135, ...  A001147
  [3] 1, 1, 4,  28,  280,   3640,   58240,   1106560, ...  A007559
  [4] 1, 1, 5,  45,  585,   9945,  208845,   5221125, ...  A007696
  [5] 1, 1, 6,  66, 1056,  22176,  576576,  17873856, ...  A008548
  [6] 1, 1, 7,  91, 1729,  43225, 1339975,  49579075, ...  A008542
  [7] 1, 1, 8, 120, 2640,  76560, 2756160, 118514880, ...  A045754
  [8] 1, 1, 9, 153, 3825, 126225, 5175225, 253586025, ...  A045755

Links

Crossrefs

Rows: A000012, A000142, A001147, A007559, A007696, A008548, A008542, A045754, A045755.
Columns: A000012, A000027, A000384, A011199, A011245.
Variant: A256268. Main diagonal: A092985.
Cf. A124320, A008279, A326323.

Programs

  • SageMath
    def A(n, k): return n**k * rising_factorial(1/n, k) if n > 0 else 1
for n in range(9): print([A(n, k) for k in range(8)])

Formula

Let rf(n, k) denote the rising factorial and ff(n,k) the falling factorial.
A(n, k) = n^k * rf(1/n, k) if n > 0 else 1.
A(n, k) = (-n)^k * ff(-1/n, k) if n > 0 else 1.
A(n, k) = (n^k * Gamma(k + 1/n)) / Gamma(1/n) for n > 0.
A(n, k) = ((-n)^k * Gamma(1 - 1/n)) / Gamma(1 - 1/n - k) for n > 0.
A(n, k) = k! * [x^k](1 - n*x)^(-1/n).
A(n, k) = [x^k] hypergeom([1, 1/n], [], n*x).
Column n + 1 has a linear recurrence with constant coefficients and signature ((-1)^k*binomial(n+1, n-k) for k=0..n).
Showing 1-7 of 7 results.

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