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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).

%I A353690 #38 Dec 26 2024 21:44:30
%S A353690 1,5,18,5,53,25,139,90,333,265,14,748,695,70,1592,1665,252,3246,3740,
%T A353690 742,6379,7960,1946,30,12152,16230,4662,150,22524,31895,10472,540,
%U A353690 40764,60760,22288,1590,72213,112620,45444,4170,125505,203820,89306,9990,55,214378,361065,170128,22440,275
%N 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).
%C A353690 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.
%C A353690 Column k is the partial sums of the k-th column of the triangle A249120.
%C A353690 Another triangle with the same row lengths and whose alternating row sums give A175254 is A262612.
%F A353690 A175254(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k).
%e A353690 Triangle begins:
%e A353690         1;
%e A353690         5;
%e A353690        18,       5;
%e A353690        53,      25;
%e A353690       139,      90;
%e A353690       333,     265,      14;
%e A353690       748,     695,      70;
%e A353690      1592,    1665,     252;
%e A353690      3246,    3740,     742;
%e A353690      6379,    7960,    1946,     30;
%e A353690     12152,   16230,    4662,    150;
%e A353690     22524,   31895,   10472,    540;
%e A353690     40764,   60760,   22288,   1590;
%e A353690     72213,  112620,   45444,   4170;
%e A353690    125505,  203820,   89306,   9990,    55;
%e A353690    214378,  361065,  170128,  22440,   275;
%e A353690    360473,  627525,  315336,  47760,   990;
%e A353690    597450, 1071890,  570696,  97380,  2915;
%e A353690    977196, 1802365, 1010982, 191370,  7645;
%e A353690   1578852, 2987250, 1757070, 364560, 18315;
%e A353690   2522157, 4885980, 3001292, 675720, 41140, 91;
%e A353690   ...
%e A353690 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).
%Y A353690 Column 1 is A353689.
%Y A353690 Row n has length A003056(n).
%Y A353690 Column k starts in row A000217(k).
%Y A353690 The first element in column k is A000330(k).
%Y A353690 Alternating row sums give A175254.
%Y A353690 Cf. A000203, A000716, A196020, A210843, A236104, A237593, A245092, A249120, A252117, A262612.
%K A353690 nonn,tabf
%O A353690 1,2
%A A353690 _Omar E. Pol_, May 04 2022