A175254 -id:A175254 - cp's OEIS frontend

A086718 Convolution of sequence of primes with sequence sigma(n).

2, 9, 22, 48, 85, 151, 231, 355, 500, 709, 937, 1267, 1617, 2069, 2575, 3193, 3860, 4686, 5549, 6593, 7725, 8985, 10337, 11961, 13591, 15464, 17498, 19714, 22036, 24690, 27378, 30382, 33603, 37023, 40597, 44733, 48720, 53152, 57950, 62978, 68074, 73898, 79558

Offset: 1

Jon Perry, Jul 29 2003

nonn

From Omar E. Pol, Dec 06 2021: (Start)
Antidiagonal sums of A272214.
Convolution of A000040 and A000203.
Convolution of A054541 and A024916.
Convolution of the nonzero terms of A007504 and A340793.
a(n) is also the volume of a tower or polycube in which the successive terraces are the symmetric representation of sigma(k), k = 1..n starting from the top, and the successive heights of the terraces are the prime numbers starting from the base. (End)

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

Cf. A000040, A000203.

Cf. A007504, A024916, A054541, A066186, A175254, A191831, A237593, A272214, A340793.

N:= 100: # to get a(1)..a(N)
P:= [seq(ithprime(i),i=1..N+1)]:
S:= [seq(numtheory:-sigma(i),i=1..N+1)]:
seq(add(P[i]*S[n-i],i=1..n-1),n=2..N+1); # Robert Israel, Sep 09 2020

p=primes(30); s=vector(30,i, sigma(i)); conv(u,v)=local(w); w=vector(length(u),i,sum(j=1,i,u[j]*v[i+1-j])); w;
conv(p,s)

More terms from Robert Israel, Sep 09 2020

A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 4, 6, 3, 7, 8, 9, 4, 6, 14, 12, 12, 5, 12, 12, 21, 16, 15, 6, 8, 24, 18, 28, 20, 18, 7, 15, 16, 36, 24, 35, 24, 21, 8, 13, 30, 24, 48, 30, 42, 28, 24, 9, 18, 26, 45, 32, 60, 36, 49, 32, 27, 10, 12, 36, 39, 60, 40, 72, 42, 56, 36, 30, 11, 28, 24, 54, 52, 75, 48, 84, 48, 63, 40, 33, 12

Offset: 1

Omar E. Pol, Sep 22 2018

			The corner of the square array begins:
         A000203 A074400 A272027 A239050 A274535 A274536 A319527 A319528
A000027:       1,      2,      3,      4,      5,      6,      7,      8, ...
A008585:       3,      6,      9,     12,     15,     18,     21,     24, ...
A008586:       4,      8,     12,     16,     20,     24,     28,     32, ...
A008589:       7,     14,     21,     28,     35,     42,     49,     56, ...
A008588:       6,     12,     18,     24,     30,     36,     42,     48, ...
A008594:      12,     24,     36,     48,     60,     72,     84,     96, ...
A008590:       8,     16,     24,     32,     40,     48,     56,     64, ...
A008597:      15,     30,     45,     60,     75,     90,    105,    120, ...
A008595:      13,     26,     39,     52,     65,     78,     91,    104, ...
A008600:      18,     36,     54,     72,     90,    108,    126,    144, ...
...

Index entries for sequences related to sigma(n)

Another version of A274824.
Antidiagonal sums give A175254.
Main diagonal gives A064987.
Row n lists the multiples of A000203(n).
Row 1 is A000027.
Initial zeros should be omitted in the following sequences related to the rows of the array:
Row 2-5: A008585, A008586, A008589, A008588.
Rows 6 and 11: A008594.
Rows 7-9: A008590, A008597, A008595.
Rows 10 and 17: A008600.
Rows 12-13: A135628, A008596.
Rows 14, 15 and 23: A008606.
Rows 16 and 25: A135631.
(Note that in the OEIS there are many other sequences that are also rows of this square array.)
Cf. A000203, A237593, A319526.

T:=Flat(List([1..12],n->List([1..n],k->k*Sigma(n-k+1))));; Print(T); # Muniru A Asiru, Jan 01 2019

with(numtheory): T:=(n,k)->k*sigma(n-k+1): seq(seq(T(n,k),k=1..n),n=1..12); # Muniru A Asiru, Jan 01 2019

Table[k DivisorSigma[1, #] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

A256532 Product of n and the sum of remainders of n mod k, for k = 1, 2, 3, ..., n.

Original entry on oeis.org

0, 0, 3, 4, 20, 18, 56, 64, 108, 130, 242, 204, 364, 434, 540, 576, 867, 846, 1216, 1220, 1470, 1694, 2254, 2040, 2575, 2912, 3375, 3472, 4379, 4140, 5177, 5344, 6072, 6698, 7630, 7128, 8621, 9424, 10491, 10320, 12177, 11928, 13975, 14432, 15255, 16468, 18941, 17952, 20286, 21000, 22899, 23608, 26765, 26568, 29095

Offset: 1

Omar E. Pol, May 03 2015

nonn

a(n) is also the volume (or the total number of unit cubes) of a polycube which is a right prism whose base is the symmetric representation of A004125(n).

Note that the union of this right prism and the irregular staircase after n-th stage described in A244580 and the irregular stepped pyramid after (n-1)-th stage described in A245092, form a hexahedron (or cube) of side length n. This comment is represented by the third formula.

			a(5) = 20 because 5 * (0 + 1 + 2 + 1) = 5 * 4 = 20.
a(6) = 18 because 6 * (0 + 0 + 0 + 2 + 1) = 6 * 3 = 18.
a(7) = 56 because 7 * (0 + 1 + 1 + 3 + 2 + 1) = 7 * 8 = 56.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000203, A000578, A004125, A024916, A143128, A175254, A236104, A236112, A237270, A237271, A237593, A244580, A245092, A256533.

Table[n*Sum[Mod[n,i],{i,2,n-1}],{n,55}] (* Ivan N. Ianakiev, May 04 2015 *)

vector(50, n, n*sum(k=1, n, n % k)) \\ Michel Marcus, May 05 2015

def A256532(n):
    s=0
    for k in range(1,n+1):
        s+=n%k
    return s*n # Indranil Ghosh, Feb 13 2017

from math import isqrt
def A256532(n): return n**3+n*((s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

a(n) = n * A004125(n).
a(n) = n^3 - A256533(n).
a(n) = n^3 - A143128(n) - A175254(n-1), n > 1.

A328366 a(n) is the surface area of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

6, 20, 40, 70, 102, 150, 194, 256, 318, 394, 462, 566, 646, 750, 858, 984, 1088, 1238, 1354, 1518, 1666, 1826, 1966, 2182, 2344, 2532, 2720, 2944, 3120, 3384, 3572, 3826, 4054, 4298, 4534, 4860, 5084, 5356, 5624, 5964, 6212, 6572, 6832, 7176, 7512, 7840, 8124, 8564, 8874, 9260, 9608, 10012

Offset: 1

Omar E. Pol, Oct 26 2019

nonn

			For n = 1 the first level of the stepped pyramid is a cube, so a(1) = 6.

Cf. A000217, A002378, A013661, A024916, A045943, A046092, A153485, A175254 (volume), A196020, A235791, A236104, A237270, A237271, A237591, A237593, A245092, A262626, A299692, A327329.

s=0;Do[s=s+4*DivisorSigma[1,n];t=2n(n+1);Print[(s/2)+t],{n,1,8000}] (* Metin Sariyar, Nov 20 2019 *)

from math import isqrt
def A328366(n): return (n*(n+1)<<1)-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 22 2023

a(n) = 4*A000217(n) + 2*A024916(n).
a(n) = 2*(A002378(n) + A327329(n)).
a(n) = 2*(A045943(n) + A153485(n)).
a(n) = A046092(n) + A327329(n).
a(n) = 2*A299692(n).
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2) + 2 = 3.644934... . - Amiram Eldar, Mar 21 2024

A332264 Partial sums of A334136.

Original entry on oeis.org

0, 3, 11, 32, 56, 116, 164, 269, 373, 535, 655, 963, 1131, 1443, 1779, 2244, 2532, 3195, 3555, 4353, 4993, 5749, 6277, 7657, 8401, 9451, 10491, 12003, 12843, 14931, 15891, 17844, 19380, 21162, 22794, 25979, 27347, 29567, 31695, 35205, 36885, 40821, 42669, 46281, 49713, 52953, 55161, 60989, 63725, 68282

Offset: 1

Omar E. Pol, Apr 19 2020

nonn

a(n) is also the volume after n-th step of the symmetric staircase described in A244580 except the volume of the base level.

			For n = 4 the volume of the first four levels of the symmetric staircase described in A244580 is 47, since the structure contains 47 cubes. The volume of the base level is 15, since the base of the structure contains 15 cubes, so a(4) = 47 - 15 = 32.

Cf. A000203, A024916, A064987, A143128, A175254, A237593, A244580, A256533, A334136.

a(n) = sum(k=1, n, (k-1)*sigma(k)); \\ Michel Marcus, Apr 19 2020

from math import isqrt
def A332264(n): return (((s:=isqrt(n))**2*(s+1)*(6-(s+1)*((s<<1)+1))>>1)+sum((q:=n//k)*(k*(q+1)*(3*k+(q<<1)+1)-3*((k<<1)+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 25 2023

a(n) = A143128(n) - A024916(n).

a(n) = A256533(n) - A175254(n). - Omar E. Pol, Apr 29 2020

A353690 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A353689 multiplied by A000330(k), and the first element of column k is in row A000217(k).

Original entry on oeis.org

1, 5, 18, 5, 53, 25, 139, 90, 333, 265, 14, 748, 695, 70, 1592, 1665, 252, 3246, 3740, 742, 6379, 7960, 1946, 30, 12152, 16230, 4662, 150, 22524, 31895, 10472, 540, 40764, 60760, 22288, 1590, 72213, 112620, 45444, 4170, 125505, 203820, 89306, 9990, 55, 214378, 361065, 170128, 22440, 275

Offset: 1

Omar E. Pol, May 04 2022

The alternating sum of the n-th row equals A175254(n), the volume of the stepped pyramid with n levels described in A245092, also the n-th term of the convolution of A000203 and A000027.
Column k is the partial sums of the k-th column of the triangle A249120.
Another triangle with the same row lengths and whose alternating row sums give A175254 is A262612.

			Triangle begins:
        1;
        5;
       18,       5;
       53,      25;
      139,      90;
      333,     265,      14;
      748,     695,      70;
     1592,    1665,     252;
     3246,    3740,     742;
     6379,    7960,    1946,     30;
    12152,   16230,    4662,    150;
    22524,   31895,   10472,    540;
    40764,   60760,   22288,   1590;
    72213,  112620,   45444,   4170;
   125505,  203820,   89306,   9990,    55;
   214378,  361065,  170128,  22440,   275;
   360473,  627525,  315336,  47760,   990;
   597450, 1071890,  570696,  97380,  2915;
   977196, 1802365, 1010982, 191370,  7645;
  1578852, 2987250, 1757070, 364560, 18315;
  2522157, 4885980, 3001292, 675720, 41140, 91;
  ...
For n = 6 we have that A175254(6) is equal to [1] + [1 + 3] + [1 + 3 + 4] + [1 + 3 + 4 + 7] + [1 + 3 + 4 + 7 + 6] + [1 + 3 + 4 + 7 + 6 + 12] = 1 + 4 + 8 + 15 + 21 + 33 = 82. On the other hand the alternating sum of the 6th row of the triangle is 333 - 265 + 14 = 82, equaling A175254(6).

Column 1 is A353689.
Row n has length A003056(n).
Column k starts in row A000217(k).
The first element in column k is A000330(k).
Alternating row sums give A175254.
Cf. A000203, A000716, A196020, A210843, A236104, A237593, A245092, A249120, A252117, A262612.

A175254(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k).

A345116 Irregular triangle T(n,k) read by rows in which row n has length the n-th triangular number A000217(n) and every column k lists the positive integers A000027, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 2, 1, 1, 1, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 8, 7, 7, 6, 6, 6, 5, 5, 5, 5

Offset: 1

Omar E. Pol, Jun 08 2021

Row n lists the terms of the n-th row of A333516 in nonincreasing order.

The sum of the divisors of the terms of the n-th row of the triangle is equal to A175254(n), equaling the volume of the stepped pyramid with n levels described in A245092.

			Triangle begins:
1;
2, 1, 1;
3, 2, 2, 1, 1, 1;
4, 3, 3, 2, 2, 2, 1, 1, 1, 1;
5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1;
6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1;
...
For n = 6 the divisors of the terms of the 6th row of triangle are:
6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1;
3, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1;
2,       1, 1, 1;
1;
The sum of these divisors is equal to A175254(6) = 82, equaling the volume of the stepped pyramid with six levels described in A245092.

Mirror of A110730.
Row lengths gives A000217, n >= 1.
Row sums gives A000292, n >= 1.
Every column gives A000027.
Cf. A000203, A175254, A237593, A245092, A333516, A340581.

A123329 Let M be the matrix defined in A111490. Sequence gives M(2,1)-M(1,2), M(2,1)+M(3,1)+M(3,2)-M(1,2)-M(1,3)-M(2,3), etc.

Original entry on oeis.org

0, 1, 3, 8, 14, 26, 39, 59, 83, 115, 148, 197, 247, 307, 376, 460, 545, 651, 758, 887, 1027, 1181, 1336, 1527, 1724, 1937, 2163, 2417, 2672, 2969, 3267, 3596, 3940, 4304, 4681, 5113, 5546, 6001, 6473, 6995, 7518, 8095, 8673, 9291, 9942, 10619, 11297, 12051

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Oct 06 2006

nonn

From Omar E. Pol, Jan 20 2021: (Start)
Conjectures:
1. Convolution of A001065 and A000027.
2. Partial sums of A153485.
3. a(n) is also the difference of volume (the difference of number of cells) between two polycubes: the stepped pyramid described in A245092 which has volume A175254(n) and the stepped pyramid that represents the n-th tetrahedral number which has volume A000292(n).
In the three conjectures assuming that here the offset is 1.
For more information about the first pyramid see A237593. (End)

Ray Chandler, Table of n, a(n) for n = 0..10000 (corrected original b-file from Michael S. Branicky missing two terms at request of Christian Krause)

Cf. A072481, A111490.

Cf. also A000027, A000292, A001065, A153485, A175254, A237593, A245092.

b:= proc(n) option remember; `if`(n=0, [0$2], (p-> p
      +[numtheory[sigma](n)-n$2]+[0, p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 21 2021

b[n_] := b[n] = If[n == 0, {0, 0}, With[{p = b[n-1]}, p +
     DivisorSigma[1, n] - n + {0, p[[1]]}]];
a[n_] := b[n+1][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 10 2021, after Alois P. Heinz *)

def M(n,k): return 1 + (k-1)%n
def a(n):
  return sum(M(i,j)-M(j,i) for i in range(2, n+3) for j in range(1, i))
print([a(n) for n in range(48)]) # Michael S. Branicky, Jan 20 2021

a(n) = binomial(n+2,3) - A072481(n+1). - Robert Israel, Aug 13 2015
a(n) = A175254(n+1) - A000292(n+1), conjectured by Omar E. Pol, Jan 20 2021
a(n) = Sum_{i=2..(n+2)} Sum_{j=1..i-1} (M(i,j)-M(j,i)). - Michael S. Branicky, Jan 20 2021

a(14) and beyond from Michael S. Branicky, Jan 20 2021

A345272 Irregular triangle read by rows T(n,k) in which row n lists in nonincreasing order all divisors of the terms of the n-th row of triangle A110730, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Jun 12 2021

Note that in the definition A110730 can be replaced with A333516 or with A345116 since these three triangles contain in every row the same terms but in distinct order.

The sum of n-th row is equal to A175254(n) equaling the volume (also the number of cubes) of the stepped pyramid with n levels described in A245092.

			Triangle begins:
1;
2, 1, 1, 1;
3, 2, 2, 1, 1, 1, 1, 1, 1;
4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...
For n = 3 the third row of A110730 is [1, 1, 1, 2, 2, 3], so the divisors of these terms in nonincreasing order are [3, 2, 2, 1, 1, 1, 1, 1, 1], the same as the third row of triangle.

Cf. A110730, A175254, A237593, A245092, A333516, A345116.

row(n) = my(v=[]); for (k=1, n, for (j=1, n-k+1, v = concat(v, divisors(k)))); vecsort(v,,4); \\ Michel Marcus, Jun 14 2021

A350637 Triangle read by rows: T(n,k) in which row n lists the first n terms of A024916 in reverse order, 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 8, 4, 1, 15, 8, 4, 1, 21, 15, 8, 4, 1, 33, 21, 15, 8, 4, 1, 41, 33, 21, 15, 8, 4, 1, 56, 41, 33, 21, 15, 8, 4, 1, 69, 56, 41, 33, 21, 15, 8, 4, 1, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1, 127, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1

Offset: 1

Omar E. Pol, Jan 09 2022

T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the stepped pyramid with n levels described in A245092 (see example).

			Triangle begins:
    1;
    4,  1;
    8,  4,  1;
   15,  8,  4,  1;
   21, 15,  8,  4,  1;
   33, 21, 15,  8,  4,  1;
   41, 33, 21, 15,  8,  4,  1;
   56, 41, 33, 21, 15,  8,  4,  1;
   69, 56, 41, 33, 21, 15,  8,  4,  1;
   87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
   99, 87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
  127, 99, 87, 69, 56, 41, 33, 21, 15,  8,  4,  1;
...
For n = 9 the lateral view and top view of the stepped pyramid described in A245092 look as shown below:
                        _
     9        1        |_|_
     8        4        |_ _|_
     7        8        |_ _|_|_
     6       15        |_ _ _| |_
     5       21        |_ _ _|_ _|_
     4       33        |_ _ _ _| | |_
     3       41        |_ _ _ _|_|_ _|_
     2       56        |_ _ _ _ _|_|_  |_
     1       69        |_ _ _ _ _|_ _|_ _|
.
   Level   Row 9         Lateral view of
     k     T(9,k)      the stepped pyramid
.
                        _ _ _ _ _ _ _ _ _
                       |_| | | | | | | | |
                       |_ _|_| | | | | | |
                       |_ _|  _|_| | | | |
                       |_ _ _|    _|_| | |
                       |_ _ _|  _|  _ _|_|
                       |_ _ _ _|  _| |
                       |_ _ _ _| |_ _|
                       |_ _ _ _ _|
                       |_ _ _ _ _|
.
                           Top view of
                       the stepped pyramid
.
For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the stepped pyramid, so T(9,1) = 69.
For n = 9 and k = 9 there is only one cubic cell in the level k = 9 (the top) of the stepped pyramid, so T(9,9) = 1.
The volume of the stepped pyramid (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000027 hence it's equal to A175254(9) = 248, equaling the sum of the 9th row of triangle.

Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Column k gives A024916 starting in row k.
Row sums give A175254.
Cf. A340423 (analog for the tower described in A221529).
Cf. A000027, A000203, A004736, A196020, A235791, A236104, A237591, A237593, A245092, A262626, A272172.

Join@@Array[Reverse@Array[Sum[#-Mod[#,m],{m,#}]&,#]&,12] (* Giorgos Kalogeropoulos, Jan 12 2022 *)

row(n) = Vecrev(vector(n, k, sum(i=1, k, k\i*i))); \\ Michel Marcus, Jan 22 2022

T(n,k) = A024916(A004736(n,k)).
T(n,k) = T(n,k) = A024916(n-k+1).
T(n,k) = Sum_{j=1..n} A272172(j,k).

cp's OEIS Frontend

A086718 Convolution of sequence of primes with sequence sigma(n).

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A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

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A256532 Product of n and the sum of remainders of n mod k, for k = 1, 2, 3, ..., n.

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A328366 a(n) is the surface area of the stepped pyramid with n levels described in A245092.

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A332264 Partial sums of A334136.

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A353690 Irregular triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A353689 multiplied by A000330(k), and the first element of column k is in row A000217(k).

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A345116 Irregular triangle T(n,k) read by rows in which row n has length the n-th triangular number A000217(n) and every column k lists the positive integers A000027, n >= 1, k >= 1.

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A123329 Let M be the matrix defined in A111490. Sequence gives M(2,1)-M(1,2), M(2,1)+M(3,1)+M(3,2)-M(1,2)-M(1,3)-M(2,3), etc.

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