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).

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

Author

Omar E. Pol, May 04 2022

Keywords

• nonn
• tabf

Comments

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.

Examples

			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).
		

Crossrefs

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.

Formula

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