A098360 -id:A098360 - 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

A098361 Multiplication table of the factorial numbers read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 6, 2, 2, 6, 24, 6, 4, 6, 24, 120, 24, 12, 12, 24, 120, 720, 120, 48, 36, 48, 120, 720, 5040, 720, 240, 144, 144, 240, 720, 5040, 40320, 5040, 1440, 720, 576, 720, 1440, 5040, 40320, 362880, 40320, 10080, 4320, 2880, 2880, 4320, 10080, 40320, 362880

Offset: 0

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

This sequence gives the variance of the 2-dimensional Polynomial Chaoses (see the Stochastic Finite Elements reference). - Stephen Crowley, Mar 28 2007
Antidiagonal sums of the array A are A003149 (row sums of the triangle T). - Roger L. Bagula, Oct 29 2008
The triangle T(n, k) = k!*(n-k)! appears as denominators in the coefficients of the Niven polynomials x^n*(1 - x)^n/n! = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!). These polynomials are used in a proof that Pi^2 (hence Pi) is irrational. See the Niven and Havil references. - Wolfdieter Lang, May 07 2018; corrected by Dimitri Papadopoulos, Nov 30 2023
The case T(n+1,k) = k!*(n-k+1)!, 1 <= k <= n+1, n >= 0 is the number of choices for forming a cluster (compact group) of k numbered items arranged in a line on a set of permutations of n numbered items arranged in a line. - Igor Victorovich Statsenko, Oct 13 2023
The numbers T(n,k) also appear in the denominators of the partial fraction expansion of 1/(x*(x+1)*...*(x+n)) = Sum_{k=0..n} (-1)^k * 1/(T(n,k)*(x+k)). - Dimitri Papadopoulos, Nov 30 2023
It follows from the previous comment that the numbers T(n,k) also appear in the denominators of the coefficients of the logarithms of the integral of 1/(x*(x+1)*...*(x+n)): c + Sum{k=0...n} (-1)^k * 1/(T(n,k)) * ln(x+k). - Colin Linzer, Dec 18 2024

			The array A(n, k) starts in row n=0 with columns k >= 0 as:
       1,      1,      2,       6,      24,      120, ...
       1,      1,      2,       6,      24,      120, ...
       2,      2,      4,      12,      48,      240, ...
       6,      6,     12,      36,     144,      720, ...
      24,     24,     48,     144,     576,     2880, ...
     120,    120,    240,     720,    2880,    14400, ...
     720,    720,   1440,    4320,   17280,    86400, ...
    5040,   5040,  10080,   30240,  120960,   604800, ...
   40320,  40320,  80640,  241920,  967680,  4838400, ...
  362880, 362880, 725760, 2177280, 8709120, 43545600, ...
  ...
The triangle T(n, k) begins:
n\k       0      1     2     3     4     5     6     7     8      9      10...
0:        1
1:        1      1
2:        2      1     2
3:        6      2     2     6
4:       24      6     4     6    24
5:      120     24    12    12    24   120
6:      720    120    48    36    48   120   720
7:     5040    720   240   144   144   240   720  5040
8:    40320   5040  1440   720   576   720  1440  5040 40320
9:   362880  40320 10080  4320  2880  2880  4320 10080 40320 362880
10: 3628800 362880 80640 30240 17280 14400 17280 30240 80640 362880 3628800
... - _Wolfdieter Lang_, May 07 2018

R. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach (Revised Edition), 2003, Ch 2.4 Table 2-2.
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.

Stefano Spezia, First 101 antidiagonals of the array, flattened
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
I. V. Statsenko, Problem on variants of cluster formation at permutations in ordered structures, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2023, pp. 7-10. In Russian.

Row sums A003149.

Cf. A003991, A098358, A098359, A098360.

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

seq(print(seq(k!*(n-k)!,k=0..n)),n=0..6); # Peter Luschny, Aug 23 2010

Table[(n+1)!*Beta[n-k+1, k+1], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Oct 29 2008 *)

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

T(n, k) = k!*(n-k)! =  n!/C(n,k), (0<=k<=n). - Peter Luschny, Aug 23 2010
Array A(n, k) = n!*k! = (k+n)!/binomial(k+n,n). - R. J. Mathar, Dec 10 2010
E.g.f. as array: 1/((1 - x)*(1 - y)). - Stefano Spezia, Jul 10 2020

A145216 Self-convolution of (1^3, 2^3, 3^3, 4^3, ... ).

Original entry on oeis.org

1, 16, 118, 560, 2003, 5888, 14988, 34176, 71445, 139216, 255970, 448240, 752999, 1220480, 1917464, 2931072, 4373097, 6384912, 9142990, 12865072, 17817019, 24320384, 32760740, 43596800, 57370365, 74717136, 96378426, 123213808

Offset: 1

Abdullahi Umar, Oct 05 2008

nonn

Row sums of A098360. - N. J. A. Sloane, May 31 2009

			a(3) = 118 because 1*(3^3) + (2^3)*(2^3) + (3^3)*1 = 118.

A. Umar, B. Yushau and B. M. Ghandi, (2006), "Patterns in convolution of two series", in Stewart, S. M., Olearski, J. E. and Thompson, D. (Eds), Proceedings of the Second Annual Conference for Middle East Teachers of Science, Mathematics and Computing (pp. 95-101). METSMaC: Abu Dhabi.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - _N. J. A. Sloane_, Mar 23 2014
A. Umar, B. Yushau and B. M. Ghandi, Convolution of two series, Australian Senior Maths Journal 21(2) (2007), 6-11.

Cf. A098360.

[(3*n^7+7*n^3-10*n)/420: n in [2..40]]; // Vincenzo Librandi, Mar 24 2014

f:=n->(3*n^7+7*n^3-10*n)/420;
[seq(f(n),n=0..50)];  # N. J. A. Sloane, Mar 23 2014

Table[Sum[(k - i)^3 (1 + i)^3, {i, 0, k - 1}], {k, 1, 35}] (* Clark Kimberling, Jun 17 2012 *)
CoefficientList[Series[(1 + 4 x + x^2)^2/(1 - x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Table[ListConvolve[Range[n]^3,Range[n]^3],{n,30}]//Flatten (* Harvey P. Dale, Sep 07 2024 *)

a(n) = C(n+2,3)*(3*n*(n+2)*(n^2+2*n+3)+16)/70.

G.f.: x*(1+4*x+x^2)^2/(1-x)^8. [Joerg Arndt, Jun 18 2012]

Name corrected by Clark Kimberling, Jun 17 2012

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

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A098358 Multiplication table of the triangular numbers read by antidiagonals.

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A098361 Multiplication table of the factorial numbers read by antidiagonals.

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A145216 Self-convolution of (1^3, 2^3, 3^3, 4^3, ... ).

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A098359 Multiplication table of the square numbers read by antidiagonals.

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