A011199 -id:A011199 - cp's OEIS frontend

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

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

Philippe Deléham, Jun 01 2015

A variant of A142589.

			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

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

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)

Flat(List([0..12], n-> List([0..n], k-> Product([0..n-k-1], j-> j*k+1) ))); # G. C. Greubel, Mar 04 2020

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

seq(seq( mul(j*k+1, j=0..n-k-1), k=0..n), n=0..12); # G. C. Greubel, Mar 04 2020

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 *)

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

[[ 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

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

A079588 a(n) = (n+1)(2n+1)(4n+1).

Original entry on oeis.org

1, 30, 135, 364, 765, 1386, 2275, 3480, 5049, 7030, 9471, 12420, 15925, 20034, 24795, 30256, 36465, 43470, 51319, 60060, 69741, 80410, 92115, 104904, 118825, 133926, 150255, 167860, 186789, 207090, 228811, 252000, 276705, 302974, 330855, 360396, 391645

Offset: 0

N. J. A. Sloane, Jan 26 2003

Apart from offset, same as A100147.

R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100147.

Cf. A258721 (first differences), A011199.

a079588 n = product $ map ((+ 1) . (* n)) [1, 2, 4]
-- Reinhard Zumkeller, Jun 08 2015

Table[(n + 1)*(2*n + 1)*(4*n + 1), {n, 0, 40}] (* Amiram Eldar, Jan 13 2021 *)
LinearRecurrence[{4,-6,4,-1},{1,30,135,364},40] (* Harvey P. Dale, Aug 01 2022 *)

Sum_{n>=0} 1/a(n) = Pi/3 (cf. Tijdeman).
G.f.: (1+26*x+21*x^2)/(1-x)^4. - L. Edson Jeffery, Mar 25 2013
Sum_{n>=0} a(n)/2^n = 308; Sum_{n>=0} (-1)^n*a(n)/2^n = -4/3. - L. Edson Jeffery, Mar 25 2013
a(n) = 8*n^3 + 14*n^2 + 7*n + 1. - Reinhard Zumkeller, Jun 08 2015
Sum_{n>=0} (-1)^n/a(n) = log(2)/3 - Pi/2 + sqrt(2)*Pi/3 + 2*sqrt(2)*arcsin(1)/3. - Amiram Eldar, Jan 13 2021
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, Jun 23 2021

A142589 Square array T(n,m) = Product_{i=0..m} (1+n*i) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 24, 15, 4, 1, 1, 120, 105, 28, 5, 1, 1, 720, 945, 280, 45, 6, 1, 1, 5040, 10395, 3640, 585, 66, 7, 1, 1, 40320, 135135, 58240, 9945, 1056, 91, 8, 1, 1, 362880, 2027025, 1106560, 208845, 22176, 1729, 120, 9, 1, 1, 3628800, 34459425, 24344320, 5221125, 576576, 43225, 2640, 153, 10, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 22 2008

Antidiagonal sums are {1, 2, 4, 11, 45, 260, 1998, 19735, 244797, 3729346, 68276276, ...}.

			The transpose of the array is:
    1,    1,     1,     1,      1,      1,      1,      1,     1,
    1,    2,     3,     4,      5,      6,      7,      8,     9,
    1,    6,    15,    28,     45,     66,     91,     120,   153, ... A000384
    1,   24,   105,   280,    585,   1056,   1729,    2640,  3825, ... A011199
    1,  120,   945,  3640,   9945,  22176,  43225,   76560, 126225,... A011245
    1,  720, 10395, 58240, 208845, 576576, 1339975, 2756160,...
        /      |       \       \
   A000142  A001147  A007559  A007696

G. C. Greubel, Antidiagonal rows n = 0..100, flattened

Cf. A000142, A006882(2n-1) = A001147, A007661(3n-2) = A007559, A007662(4n-3) = A007696, A153274.

function T(n,k)
  if k eq 0 or n eq 0 then return 1;
  else return (&*[j*k+1: j in [0..n]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 05 2020

T:= (n, k)-> `if`(n=0, 1, mul(j*k+1, j=0..n)):
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Mar 05 2020

T[n_, k_]= If[n==0, 1, Product[1 + k*i, {i,0,n}]]; Table[T[n-k, k], {n,0,10}, {k,0,n}]//Flatten

T(n, k) = if(n==0, 1, prod(j=0, n, j*k+1) );
for(n=0, 12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Mar 05 2020

def T(n, k):
    if (k==0 and n==0): return 1
    else: return product(j*k+1 for j in (0..n))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 05 2020

Edited by M. F. Hasler, Oct 28 2014

More terms added by G. C. Greubel, Mar 05 2020

A162651 Numbers which can be expressed as the product of 3 positive integers in arithmetic progression.

Original entry on oeis.org

1, 6, 8, 15, 24, 27, 28, 45, 48, 60, 64, 66, 80, 91, 105, 120, 125, 153, 162, 168, 190, 192, 210, 216, 224, 231, 276, 280, 288, 312, 315, 325, 336, 343, 360, 378, 384, 405, 435, 440, 480, 496, 504, 510, 512, 528, 561, 585, 624, 627, 630, 640, 648, 693, 703, 720

Offset: 1

Franklin T. Adams-Watters, Jul 08 2009

nonn

Numbers of the form i*(i+j)*(i+2j), where i > 0 and j >= 0.

			1 = 1*1*1, 6 = 1*2*3, 8 = 2*2*2, 15 = 1*3*5, 24 = 2*3*4.
120 = 1*8*15 = 2*6*10 = 4*5*6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000578, A007531, A000384, A033996, A011199.

N:= 1000: # for all terms <= N
S:= {}:
for i from 1 to floor(N^(1/3)) do
  S:= S union {seq(i*(i+j)*(i+2*j),j=0..floor((sqrt(i^4 + 8*i*N)-3*i^2)/(4*i)))}
od:
A:= sort(convert(S,list)); # Robert Israel, Feb 05 2020

al(n)={local(v,inc,prd);
v=vector(n);inc=[0];prd=[1];
for(k=1,n,
v[k]=vecmin(prd);
if(v[k]==prd[ #prd],inc=concat(inc,[0]);prd=concat(prd,[(#inc)^3]));
for(j=1,#prd,if(prd[j]==v[k],inc[j]++;prd[j]=j*(j+inc[j])*(j+2*inc[j]))));
v}

from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A162651_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        for r in divisors(m,generator=True):
            if is_square(r**2-m//r):
                yield m
                break
A162651_list = list(islice(A162651_gen(),20)) # Chai Wah Wu, Jul 03 2023

A111670 Array T(n,k) read by antidiagonals: the k-th column contains the first column of the k-th power of A039755.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 24, 1, 1, 5, 28, 105, 116, 1, 1, 6, 45, 280, 929, 648, 1, 1, 7, 66, 585, 3600, 9851, 4088, 1, 1, 8, 91, 1056, 9865, 56240, 121071, 28640, 1

Offset: 1

Gary W. Adamson, Aug 14 2005

			 1     1       1          1          1          1          1          1
 1     2       3          4          5          6          7          8
 1     6      15         28         45         66         91        120
 1    24     105        280        585       1056       1729       2640
 1   116     929       3600       9865      22036      43001      76224
 1   648    9851      56240     203565     565096    1318023    2717856
 1  4088  121071    1029920    4953205   17148936   47920803  115146816
 1 28640 1685585   21569600  138529105  600001696 2012844225 5644055040

Cf. A039755, A007405 (column 2), A000384 (row 2), A011199 (row 3).

A111670 := proc(n,k)
    local A,i,j ;
    A := Matrix(n,n) ;
    for i from 1 to n do
    for j from 1 to n do
        A[i,j] := A039755(i-1,j-1) ;
    end do:
    end do:
    LinearAlgebra[MatrixPower](A,k) ;
    %[n,1] ;
end proc:
for d from 2 to 12 do
    for n from  1 to d-1 do
        printf("%d,",A111670(n,d-n)) ;
    end do:
end do: # R. J. Mathar, Jan 27 2023

nmax = 10;
A[n_, k_] := Sum[(-1)^(k-j)*(2j+1)^n*Binomial[k, j], {j, 0, k}]/(2^k*k!);
A039755 = Array[A, {nmax, nmax}, {0, 0}];
T = Table[MatrixPower[A039755, n][[All, 1]], {n, 1, nmax}] // Transpose;
Table[T[[n-k+1, k]], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 02 2024 *)

Let A039755 (an analog of Stirling numbers of the second kind) be an infinite lower triangular matrix M; then the vector M^k * [1, 0, 0, 0, ...] (first column of the k-th power) is the k-th column of this array.

Definition simplified by R. J. Mathar, Jan 27 2023

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

Peter Luschny, Dec 18 2023

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

			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

Index entries for sequences related to factorial numbers.

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.

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)])

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).

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

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A079588 a(n) = (n+1)*(2*n+1)*(4*n+1).

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A142589 Square array T(n,m) = Product_{i=0..m} (1+n*i) read by antidiagonals.

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A162651 Numbers which can be expressed as the product of 3 positive integers in arithmetic progression.

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A111670 Array T(n,k) read by antidiagonals: the k-th column contains the first column of the k-th power of A039755.

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

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A079588 a(n) = (n+1)(2n+1)(4n+1).