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A347026 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.

%I A347026 #77 Aug 31 2024 17:26:14
%S A347026 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 A347026 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 A347026 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 A347026 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 A347026 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 A347026 Sums of the rising diagonals yield sequence A007980.
%C A347026 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 A347026 Harvey P. Dale, <a href="/A347026/b347026.txt">Table of n, a(n) for n = 1..1000</a>
%H A347026 Eddie Gutierrez, <a href="http://oddwheel.com/Mathx.html">A Novel Method for Multiplying Large Numbers</a>.
%H A347026 Eddie Gutierrez, <a href="http://oddwheel.com/CAT.html">The Column Addition Triangle (CAT)-A Pascal Analog (Part I)</a>.
%H A347026 Eddie Gutierrez, <a href="http://oddwheel.com/CAT2.html">The Column Addition Triangle (CAT) and Polynomials</a>.
%F A347026 T(n,k) = 2k - 1      for   1 <= k <= n,
%F A347026          4n - 2k + 1 for n+1 <= k <= 2n.
%e A347026 Triangle begins:
%e A347026   1,  1;
%e A347026   1,  3,  3,  1;
%e A347026   1,  3,  5,  5,  3,  1;
%e A347026   1,  3,  5,  7,  7,  5,  3,  1;
%e A347026   1,  3,  5,  7,  9,  9,  7,  5,  3,  1;
%e A347026   1,  3,  5,  7,  9, 11, 11,  9,  7,  5,  3,  1;
%e A347026   1,  3,  5,  7,  9, 11, 13, 13, 11,  9,  7,  5,  3,  1;
%e A347026   1,  3,  5,  7,  9, 11, 13, 15, 15, 13, 11,  9,  7,  5,  3,  1;
%e A347026 ...
%t A347026 Array[Join[#, Reverse[#]] &@Range[1, 2 # - 1, 2] &, 9] // Flatten (* _Michael De Vlieger_, Aug 18 2021 *)
%t A347026 Flatten[Table[Join[Range[1,2n+1,2],Range[2n+1,1,-2]],{n,0,10}]] (* _Harvey P. Dale_, Aug 31 2024 *)
%o A347026 (C)
%o A347026 #include <stdio.h>
%o A347026 int main()
%o A347026 {
%o A347026    int n, k;
%o A347026    for (n=1; n<=13; n++)
%o A347026    {
%o A347026       for (k=1; k<=n; k++)
%o A347026       {
%o A347026          printf("%d ", 2*k - 1);
%o A347026       }
%o A347026       for (k=n+1; k<=2*n; k++)
%o A347026       {
%o A347026          printf("%d ", 4*n - 2*k + 1);
%o A347026       }
%o A347026       printf("\n");
%o A347026    }
%o A347026    return 0;
%o A347026 }
%o A347026 (PARI) row(n) = n*=2; vector(n, k, min(2*k-1, 2*(n-k)+1)); \\ _Michel Marcus_, Aug 17 2021
%Y A347026 Even-indexed rows of A157454.
%Y A347026 Antidiagonal sums give A007980.
%Y A347026 Row lengths give nonzero terms of A005843.
%Y A347026 Cf. A004737.
%K A347026 nonn,tabf
%O A347026 1,4
%A A347026 _Eddie Gutierrez_, Aug 11 2021
%E A347026 Better definition from _Omar E. Pol_, Aug 14 2021