%I #16 May 08 2021 08:32:15

%S 1,1,1,1,2,1,1,3,3,1,1,4,5,5,1,1,5,7,9,9,1,1,6,9,13,17,17,1,1,7,11,17,

%T 25,33,33,1,1,8,13,21,33,49,65,65,1,1,9,15,25,41,65,97,129,129,1,1,10,

%U 17,29,49,81,129,193,257,257,1,1,11,19,33,57,97,161,257,385,513,513,1

%N Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

%D G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

%H Andrew Howroyd, <a href="/A046688/b046688.txt">Table of n, a(n) for n = 0..1325</a>

%F A(m,n) = 1 + n*2^(m-1) for m > 1. - _Andrew Howroyd_, Mar 07 2020

%F As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - _Gus Wiseman_, May 08 2021

%e From _Gus Wiseman_, May 08 2021: (Start):

%e Array A(m,n) = 1 + n*2^(m-1) begins:

%e n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:

%e m=0: 1 1 1 1 1 1 1 1 1 1

%e m=1: 1 2 3 5 9 17 33 65 129 257

%e m=2: 1 3 5 9 17 33 65 129 257 513

%e m=3: 1 4 7 13 25 49 97 193 385 769

%e m=4: 1 5 9 17 33 65 129 257 513 1025

%e m=5: 1 6 11 21 41 81 161 321 641 1281

%e m=6: 1 7 13 25 49 97 193 385 769 1537

%e m=7: 1 8 15 29 57 113 225 449 897 1793

%e m=8: 1 9 17 33 65 129 257 513 1025 2049

%e m=9: 1 10 19 37 73 145 289 577 1153 2305

%e Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:

%e 1

%e 1 1

%e 1 2 1

%e 1 3 3 1

%e 1 4 5 5 1

%e 1 5 7 9 9 1

%e 1 6 9 13 17 17 1

%e 1 7 11 17 25 33 33 1

%e 1 8 13 21 33 49 65 65 1

%e 1 9 15 25 41 65 97 129 129 1

%e 1 10 17 29 49 81 129 193 257 257 1

%e 1 11 19 33 57 97 161 257 385 513 513 1

%e (End)

%t Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, _Gus Wiseman_, May 08 2021 *)

%t Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, _Gus Wiseman_, May 08 2021 *)

%o (PARI) A(m,n)={if(m>0, 1+n*2^(m-1), 1)}

%o { for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ _Andrew Howroyd_, Mar 07 2020

%Y Row sums are A000079.

%Y Diagonal n = m + 1 of the array is A002064.

%Y Diagonal n = m of the array is A005183.

%Y Column m = 1 of the array is A094373.

%Y Diagonal n = m - 1 of the array is A131056.

%Y A002109 gives hyperfactorials (sigma: A260146, omega: A303281).

%Y A009998(k,n) = n^k.

%Y A009999(n,k) = n^k.

%Y A057156 = (2^n)^(2^n).

%Y A062319 counts divisors of n^n.

%Y Cf. A000169, A000272, A000312, A036289, A343656, A343658.

%K nonn,tabl,easy

%O 0,5

%A _N. J. A. Sloane_

%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000