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%I #72 Sep 29 2021 13:10:32
%S 1,1,1,3,3,1,1,3,5,5,3,1,1,3,5,7,7,5,3,1,1,3,5,7,9,9,7,5,3,1,1,3,5,7,
%T 9,11,11,9,7,5,3,1,1,3,5,7,9,11,13,13,11,9,7,5,3,1,1,3,5,7,9,11,13,15,
%U 15,13,11,9,7,5,3,1,1,3,5,7,9,11,13,15,17,17,15,13,11,9,7,5,3,1
%N Irregular triangle read by rows in which row n lists the first n odd numbers, followed by the first n odd numbers in decreasing order.
%C The terms of this sequence are the numbers in an irregular triangle corresponding to the addition of rows when multiplying two large numbers via a novel method (see Links).
%C Sums of the rising diagonals yield sequence A007980.
%C When the 2n terms in row n are used as the coefficients of a (2n-1)st-order polynomial in x, dividing that polynomial by x+1 produces a (2n-2)nd-order polynomial whose coefficients are the n-th row of A004737 (if that sequence is taken as an irregular triangle with 2n-1 terms in its n-th row). E.g., for n=3, (x^5 + 3x^4 + 5x^3 + 5x^2 + 3x + 1)/(x+1) = x^4 + 2x^3 + 3x^2 + 2x + 1.
%H Eddie Gutierrez, <a href="http://oddwheel.com/Mathx.html">A Novel Method for Multiplying Large Numbers</a>.
%H Eddie Gutierrez, <a href="http://oddwheel.com/CAT.html">The Column Addition Triangle (CAT)-A Pascal Analog (Part I)</a>.
%H Eddie Gutierrez, <a href="http://oddwheel.com/CAT2.html">The Column Addition Triangle (CAT) and Polynomials</a>.
%F T(n,k) = 2k - 1 for 1 <= k <= n,
%F 4n - 2k + 1 for n+1 <= k <= 2n.
%e Triangle begins:
%e 1, 1;
%e 1, 3, 3, 1;
%e 1, 3, 5, 5, 3, 1;
%e 1, 3, 5, 7, 7, 5, 3, 1;
%e 1, 3, 5, 7, 9, 9, 7, 5, 3, 1;
%e 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1;
%e 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1;
%e 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1;
%e ...
%t Array[Join[#, Reverse[#]] &@Range[1, 2 # - 1, 2] &, 9] // Flatten (* _Michael De Vlieger_, Aug 18 2021 *)
%o (C)
%o #include <stdio.h>
%o int main()
%o {
%o int n, k;
%o for (n=1; n<=13; n++)
%o {
%o for (k=1; k<=n; k++)
%o {
%o printf("%d ", 2*k - 1);
%o }
%o for (k=n+1; k<=2*n; k++)
%o {
%o printf("%d ", 4*n - 2*k + 1);
%o }
%o printf("\n");
%o }
%o return 0;
%o }
%o (PARI) row(n) = n*=2; vector(n, k, min(2*k-1, 2*(n-k)+1)); \\ _Michel Marcus_, Aug 17 2021
%Y Even-indexed rows of A157454.
%Y Antidiagonal sums give A007980.
%Y Row lengths give nonzero terms of A005843.
%Y Cf. A004737.
%K nonn,tabf
%O 1,4
%A _Eddie Gutierrez_, Aug 11 2021
%E Better definition from _Omar E. Pol_, Aug 14 2021
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