A098361 -id:A098361 - cp's OEIS frontend

A098358 Multiplication table of the triangular numbers read by antidiagonals.

1, 3, 3, 6, 9, 6, 10, 18, 18, 10, 15, 30, 36, 30, 15, 21, 45, 60, 60, 45, 21, 28, 63, 90, 100, 90, 63, 28, 36, 84, 126, 150, 150, 126, 84, 36, 45, 108, 168, 210, 225, 210, 168, 108, 45, 55, 135, 216, 280, 315, 315, 280, 216, 135, 55, 66, 165, 270, 360, 420, 441, 420, 360

Offset: 0

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

The number of rectangles to be found in a grid of size X by y. For example a(2, 2) = 9 since a 2 x 2 grid contains one rectangle of size 2 X 2, 4 of size 1 X 2 and 4 of size 1 X 1. - Hugo van der Sanden, May 24 2007

			Triangle begins:
   1;
   3,  3;
   6,  9,  6;
  10, 18, 18, 10;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A003991, A098359, A098360, A098361.

a[n_, k_] := Binomial[k + 1, 2]*Binomial[n + 1, 2]; Table[a[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 22 2017 *)

a(m,n) = m*(m+1)*n*(n+1)/4.

G.f.: x*y / ((1-x)^3 * (1-y)^3). - Ralf Stephan, Oct 27 2004

A098360 Multiplication table of the cube numbers read by antidiagonals.

Original entry on oeis.org

1, 8, 8, 27, 64, 27, 64, 216, 216, 64, 125, 512, 729, 512, 125, 216, 1000, 1728, 1728, 1000, 216, 343, 1728, 3375, 4096, 3375, 1728, 343, 512, 2744, 5832, 8000, 8000, 5832, 2744, 512, 729, 4096, 9261, 13824, 15625, 13824, 9261, 4096, 729, 1000, 5832, 13824

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

			1; 8,8; 27,64,27; 64,216,216,64; ...

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)

Cf. A003991, A098358, A098359, A098361.

Row sums: A145216. - N. J. A. Sloane, May 31 2009

Flat(List([2..11],m->List([1..m-1],i->i^3*(m-i)^3))); # Muniru A Asiru, Jun 27 2018

seq(seq(i^3*(m-i)^3,i=1..m-1),m=2..10); # Robert Israel, Jun 27 2018

With[{s = Range[10]^3}, Table[s[[#]] s[[j]] &[i - j + 1], {i, Length@s}, {j, i}]] // Flatten (* Michael De Vlieger, Jun 27 2018 *)

G.f. as rectangular array: [xy(1+4x+x^2)(1+4y+y^2)] / [(1-x)^4 * (1-y)^4 ]. - Ralf Stephan, Oct 27 2004, corrected by Robert Israel, Jun 27 2018

a(n) = A003991(n)^3.- Robert Israel, Jun 27 2018

More terms from Ralf Stephan, Oct 27 2004

Offset corrected by Robert Israel, Jun 27 2018

A143216 Triangle read by rows: T(n,k) = n!*k!, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 6, 12, 36, 24, 24, 48, 144, 576, 120, 120, 240, 720, 2880, 14400, 720, 720, 1440, 4320, 17280, 86400, 518400, 5040, 5040, 10080, 30240, 120960, 604800, 3628800, 25401600, 40320, 40320, 80640, 241920, 967680, 4838400, 29030400, 203212800, 1625702400

Offset: 0

Gary W. Adamson, Jul 30 2008

			First few rows of the triangle =
    1;
    1,   1;
    2,   2,    4;
    6,   6,   12,   36;
   24,  24,   48,  144,   576;
  120, 120,  240,  720,  2880, 14400;
  720, 720, 1440, 4320, 17280, 86400, 518400;
  ...
T(6,3) = 4320 = 6!*3! = 720*6.

Stefano Spezia, First 101 rows of the triangle, flattened

Cf. A000142, A098361 (as an array), A143217 (row sums).

F:=Factorial; [F(n)*F(k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2022

Table[n!k!,{n,0,8},{k,0,n}] (* Stefano Spezia, Jul 09 2020 *)

f=factorial; flatten([[f(n)*f(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 12 2022

T(n,k) = n!*k!, 0 <= k <= n.

E.g.f.: 1/((1 - x)*(1 - y)). - Stefano Spezia, Jul 09 2020

A098359 Multiplication table of the square numbers read by antidiagonals.

Original entry on oeis.org

1, 4, 4, 9, 16, 9, 16, 36, 36, 16, 25, 64, 81, 64, 25, 36, 100, 144, 144, 100, 36, 49, 144, 225, 256, 225, 144, 49, 64, 196, 324, 400, 400, 324, 196, 64, 81, 256, 441, 576, 625, 576, 441, 256, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 400, 729, 1024, 1225, 1296, 1225, 1024, 729, 400, 121

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k-1) = n. See A003991. - Charlie Marion, Apr 22 2013

			Square array A(n,k) begins:
   1,   4,   9,  16,   25,   36,   49, ...
   4,  16,  36,  64,  100,  144,  196, ...
   9,  36,  81, 144,  225,  324,  441, ...
  16,  64, 144, 256,  400,  576,  784, ...
  25, 100, 225, 400,  625,  900, 1225, ...
  36, 144, 324, 576,  900, 1296, 1764, ...
  49, 196, 441, 784, 1225, 1764, 2401, ...

Cf. A000290, A003991, A098358, A098360, A098361, A213547.
Antidiagonal sums give A033455.
Main diagonal gives A000583.

A:= (n,k)-> (n*k)^2:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, May 19 2025

A(n,k) = n^2*k^2.
G.f.: [xy(1+x)(1+y)] / [(1-x)^3 * (1-y)^3 ]. - Ralf Stephan, Oct 27 2004
Sum_{j=1..n} A(j,1+n-j)*j = A213547(n). - Alois P. Heinz, May 19 2025

Offset corrected by Alois P. Heinz, May 19 2025

A356545 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of first order representing the Bernoulli numbers as B_n = p_n(1) / (n + 1)!.

Original entry on oeis.org

1, 1, 0, 2, -1, 0, 6, -8, 2, 0, 24, -66, 44, -6, 0, 120, -624, 792, -312, 24, 0, 720, -6840, 14496, -10872, 2736, -120, 0, 5040, -86400, 285840, -347904, 171504, -28800, 720, 0, 40320, -1244880, 6181920, -11245680, 8996544, -3090960, 355680, -5040, 0

Offset: 0

Peter Luschny, Aug 11 2022

The Bernoulli numbers with B(1) = 1/2 can be represented as the weighted sum of Eulerian numbers, where we use the definition as given by Graham et al., Eulerian(n, k) = A173018(n, k). For n >= 0 we have
   B_(n) = (1/(n + 1)) * Sum_{k=0..n} (-1)^k * Eulerian(n, k) / binomial(n, k).
The formula was given by Worpitsky in 1883 (see link) as an example for the application of a formula of Schlömilch from 1856. In 2019 the identity was proved in the modern fashion by Gessel on MathOverflow.
For a variant of this identity see the first formula in A356546.
An analogous representation based on the Eulerian numbers of second order is given in A356547.

			The table T(n, k) of the coefficients, sorted in ascending order, starts:
[0]     1;
[1]     1,        0;
[2]     2,       -1,       0;
[3]     6,       -8,       2,         0;
[4]    24,      -66,      44,        -6,       0;
[5]   120,     -624,     792,      -312,      24,        0;
[6]   720,    -6840,   14496,    -10872,    2736,     -120,      0;
[7]  5040,   -86400,  285840,   -347904,  171504,   -28800,    720,     0;
[8] 40320, -1244880, 6181920, -11245680, 8996544, -3090960, 355680, -5040, 0;

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 268. (Since the thirty-fourth printing, Jan. 2022, with B(1) = 1/2.)

Ira Gessel, Eulerian number identity, MathOverflow, Apr 2019.
Peter Luschny, How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?, MathOverflow, Feb. 2021.
Peter Luschny, Eulerian polynomials.
Oskar Schlömilch, Ueber die Bernoulli'sche Funktion und deren Gebrauch bei der Entwickelung halbconvergenter Reihen, Zeitschrift fuer Mathematik und Pysik, vol. 1 (1856), p. 193-211.
Julius Worpitsky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik (Crelle), 94 (1883), 203-232. See page 22, first formula.

Cf. A173018 (Eulerian number), A164555(n)/A027642(n) (Bernoulli numbers with B(1) = 1/2), A129814 (row sums, but different sign for n = 1).

Cf. A098361, A123125, A356546, A356547, A356601, A356602.

E1 := proc(n, k) combinat:-eulerian1(n, k) end:
p := (n, x) -> add(E1(n, k)*k!*(n - k)!*(-x)^k, k = 0..n):
seq(print(seq(coeff(p(n, x), x, k), k=0..n)), n = 0..8);
seq(p(n, 1)/(n + 1)!, n = 0..14); # check the Bernoulli representation

T[n_, k_] := k! * (n-k)! * Sum[(-1)^(k-j) * (k-j+1)^n * Binomial[n+1, j], {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // TableForm
(* Diagonals: *)
d[n_, k_] := k! * (n - k)! * Sum[(-1)^(n-k-j)*(n - j - k + 1)^n * Binomial[n + 1, j], {j, 0, n - k}];

Let p_n(x) = Sum_{k=0..n} Eulerian(n, k)*k!*(n - k)! * (-x)^k. For x = 1 these polynomials give rise to the representation Bernoulli(n) = p_n(1) / (n + 1)!.
T(n, k) = [x^k] p_n(x).
T(n, k) = (-1)^k*Eulerian(n, k)*k!*(n - k)!.
T(n, k) = k! * (n-k)! * Sum_{j=0..k} (-1)^(k-j)*(k-j+1)^n*binomial(n+1, j).
T(n, k) = (-1)^k * A173018(n, k) * A098361(n, k).
T(n, k) = (-1)^k * A123125(n, n - k) * A098361(n, n - k).

A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.

Original entry on oeis.org

1, 1, -2, 1, -6, 12, 1, -12, 60, -120, 1, -20, 180, -840, 1680, 1, -30, 420, -3360, 15120, -30240, 1, -42, 840, -10080, 75600, -332640, 665280, 1, -56, 1512, -25200, 277200, -1995840, 8648640, -17297280, 1, -72, 2520, -55440, 831600, -8648640, 60540480, -259459200, 518918400, 1, -90, 3960, -110880, 2162160, -30270240, 302702400, -2075673600, 8821612800, -17643225600, 1, -110, 5940, -205920, 5045040, -90810720, 1210809600, -11762150400, 79394515200, -335221286400, 670442572800

Offset: 0

Wolfdieter Lang, May 07 2018

The Niven potentials N(n, x) = (1/n!)*x^n*(1 - x)^n = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!), with (n-k)!*k! = A098361(n, k), are used to prove the irrationality of Pi^2 (hence Pi). See the Niven and Havil references.

The row polynomials R(n, x) = Sum_{k=0..n} T(n, k) *x^k are R(n, x) = y_n(-2*x), with the Bessel polynomials of Krall and Frink y_n(x) with coefficients given in A001498. There the references are given. - Wolfdieter Lang, May 12 2018

			The triangle T(n, k) begins:
n\k 0   1    2      3      4        5        6          7         8 ...
0:  1
1:  1  -2
2:  1  -6   12
3:  1 -12   60   -120
4:  1 -20  180   -840   1680
5:  1 -30  420  -3360  15120   -30240
6:  1 -42  840 -10080  75600  -332640    66528
7:  1 -56 1512 -25200 277200 -1995840  8648640  -17297280
8:  1 -72 2520 -55440 831600 -8648640 60540480 -259459200 518918400
...

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.

Muniru A Asiru, Rows n = 0..50

Row sums are A002119.

Cf. A098361, A001498, A113025, A113216.

Flat(List([0..10],n->List([0..n],k->(-1)^k*Binomial(n,n-k)*Factorial(n+k)/Factorial(n)))); # Muniru A Asiru, May 15 2018

T := (n, k) -> (-1)^k*pochhammer(n+1, k)*binomial(n, k):
seq(print(seq(T(n, k), k=0..n)), n=0..9); # Peter Luschny, May 11 2018

T(n,k)=(-1)^k*binomial(n,n-k)*binomial(n+k,n)*k! \\ M. F. Hasler, May 09 2018

T(n, k) = (d/dx)^(n+k) N(n, x) |_{x=0} =: N^{(n+k)}(n, 0), with N(n, x) = (1/n!)*x^n*(1 - x)^n, for n >= 0, k = 0..n.
N^{(n+k)}(n, 1) = (-1)^(n+k)*T(n, k), which has for even n the unsigned rows, and for odd n the unsigned row entries with negative signs.
T(n, k) = (-1)^k*binomial(n, n-k)*((n+k)!/n!).
T(n, k) = (-1)^k*A113025(n,k) with A113025(n,k) = (n+k)!/(k!*(n-k)!) = abs(A113216(n,k)). - M. F. Hasler, May 09 2018
T(n, k) = (-1)^k*Pochhammer(n+1, k)*binomial(n, k). - Peter Luschny, May 11 2018
Recurrence: from the one of the row polynomials R(n, x) = y_n(-2*x): R(n, x) = -2*(2*n-1)*x*R(n-1, x) + R(n-2, x), with R(-1, x) = 1 = R(0, x) = 1, n >= 1 (see A001498), this becomes, for n >= 0, k = 0..n:
  T(n, k) = 0 for n < k, T(n, -1) = 0, T(0, 0) = 1 = T(1, 0) and otherwise
  T(n, k) = -2*(2*n-1)*T(n-1, k-1) + T(n-2, k). - Wolfdieter Lang, May 12 2018

A349424 Integers that are the product of two factorials A000142.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 120, 144, 240, 576, 720, 1440, 2880, 4320, 5040, 10080, 14400, 17280, 30240, 40320, 80640, 86400, 120960, 241920, 362880, 518400, 604800, 725760, 967680, 2177280, 3628800, 4838400, 7257600, 8709120, 21772800, 25401600, 29030400

Offset: 1

Frank M Jackson, Nov 17 2021

nonn

The first term a(1) = 0!*0! = 1!*0! = 1!*1!.

This is also the number of perfect matchings in a graph connecting C-G and A-U pairs in a RNA string (see Rosalind link). - Lorenzo Pieri, Mar 02 2023

			720 is a term since 720 = 5!*3! = 6!*0! = 6!*1!.

Rosalind, Perfect Matchings and RNA Secondary Structures.

Cf. A000142, A098361.

Subsequence of A001013.

(Union@Flatten@Table[x!*y!, {x, 0, 100}, {y, 0, x}])[[1 ;; 50]]

lista(nn) = select(x->(x<2*nn!), setbinop((x,y)->x!*y!, [1..nn])); \\ Michel Marcus, Nov 17 2021

cp's OEIS Frontend

A098358 Multiplication table of the triangular numbers read by antidiagonals.

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A098360 Multiplication table of the cube numbers read by antidiagonals.

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A143216 Triangle read by rows: T(n,k) = n!*k!, 0 <= k <= n.

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A098359 Multiplication table of the square numbers read by antidiagonals.

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A356545 Triangle read by rows. T(n, k) are the coefficients of polynomials p_n(x) based on the Eulerian numbers of first order representing the Bernoulli numbers as B_n = p_n(1) / (n + 1)!.

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A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.

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A349424 Integers that are the product of two factorials A000142.

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