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A288870 Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.

%I A288870 #15 Jan 28 2025 07:45:38
%S A288870 2,3,4,5,7,6,9,13,11,8,17,25,21,15,10,33,49,41,29,19,12,65,97,81,57,
%T A288870 37,23,14,129,193,161,113,73,45,27,16,257,385,321,225,145,89,53,31,18,
%U A288870 513,769,641,449,289,177,105,61,35,20,1025,1537,1281,897,577,353,209,121,69,39,22
%N A288870 Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.
%C A288870 This entry was motivated by a class work of Ferran D.
%F A288870 Array A(k, n) = (2*k+1)*2^n + 1 for k >= 0 and n >= 0.
%F A288870 Triangle T(m, k) = A(k, m-k) = (2*k+1)*2^(m-k) + 1, k >= m >= 0, otherwise T(m, k) = 0.
%F A288870 O.g.f. for column k of T: x^k*(2*(k+1) - (2*k+3)*x)/((1-2*x)*(1-x)), k >= 0.
%F A288870 E.g.f. for column k of T (without leading 0's): (2*k+1)*exp(2*x) + exp(x), k>=0.
%F A288870 E.g.f. for column k of T: 2^(-k)*(2*k+1)*exp(2*x) + exp(x) - S(k,x), with S(k, x) = 2^(-k)* Sum_{m=1..k} A288871(k,m)*x^(m-1)/(m-1)! if k >=1 and S(0,x) = 0.
%e A288870 The array A begins:
%e A288870 k\n  0  1  2   3   4   5    6    7    8    9    10 ...
%e A288870 0:   2  3  5   9  17  33   65  129  257  513  1025
%e A288870 1:   4  7 13  25  49  97  193  385  769 1537  3073
%e A288870 2:   6 11 21  41  81 161  321  641 1281 2561  5121
%e A288870 3:   8 15 29  57 113 225  449  897 1793 3585  7169
%e A288870 4:  10 19 37  73 145 289  577 1153 2305 4609  9217
%e A288870 5:  12 23 45  89 177 353  705 1409 2817 5633 11265
%e A288870 6:  14 27 53 105 209 417  833 1665 3329 6657 13313
%e A288870 7:  16 31 61 121 241 481  961 1921 3841 7681 15361
%e A288870 8:  18 35 69 137 273 545 1089 2177 4353 8705 17409
%e A288870 9:  20 39 77 153 305 609 1217 2433 4865 9729 19457
%e A288870 ...
%e A288870 The triangle T begins:
%e A288870 m\k    0    1    2   3   4   5   6   7  8  9 10 ...
%e A288870 0:     2
%e A288870 1:     3    4
%e A288870 2:     5    7    6
%e A288870 3:     9   13   11   8
%e A288870 4:    17   25   21  15  10
%e A288870 5:    33   49   41  29  19  12
%e A288870 6:    65   97   81  57  37  23  14
%e A288870 7:   129  193  161 113  73  45  27 16
%e A288870 8:   257  385  321 225 145  89  53 31 18
%e A288870 9:   513  769  641 449 289 177 105 61 35 20
%e A288870 10: 1025 1537 1281 897 577 353 209 121 69 39 22
%e A288870 ...
%t A288870 Table[(2 k + 1)*2^(m - k) + 1, {m, 0, 10}, {k, 0, m}] // Flatten (* _Michael De Vlieger_, Jun 25 2017 *)
%o A288870 (PARI) A(n, k) = (2*n + 1)*2^k + 1;
%o A288870 for(n=0, 10, for(k=0, n, print1(A(k, n - k),", "))) \\ _Indranil Ghosh_, Jun 22 2017
%Y A288870 Cf. A288871. Columns of T (no 0's, or rows of A): A000051, A181565, A083575, A083686, A083705, A083683, A168596.
%Y A288870 Row sums give A077802(n+1) or A095151(n+1).
%K A288870 nonn,tabl,easy
%O A288870 0,1
%A A288870 _Wolfdieter Lang_, Jun 21 2017