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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).
2
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
1,2
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.
FORMULA
A175254(n) = Sum_{k=1..A003056(n))} (-1)^(k-1)*T(n,k).
EXAMPLE
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.
Sequence in context: A351890 A097491 A120087 * A180135 A260080 A178365
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, May 04 2022
STATUS
approved