A171840 - OEIS

%I #21 Feb 13 2022 09:28:29

%S 1,1,2,1,2,5,1,2,4,15,1,2,4,9,52,1,2,4,8,23,203,1,2,4,8,17,65,877,1,2,

%T 4,8,16,40,199,4140,1,2,4,8,16,33,104,654,21147,1,2,4,8,16,32,73,291,

%U 2296,115975,1,2,4,8,16,32,65,177,857,8569,678570

%N Triangle read by rows, truncated columns of an array formed by taking sets of P(n) = Pascal's triangle, with the 1's column shifted up n = 1,2,3,... times. Then the n-th row of the array = lim_{k->infinity}, k=1,2,3,...; (P(n))^k, deleting the first 1.

%C Row sums = A171841: (1, 3, 8, 22, 68, 241, 974, ...).

%C Right border = the Bell sequence A000110 starting (1, 2, 5, 15, 52, ...).

%C Row 2 of the array = A007476 starting (1, 1, 2, 4, 9, 23, 65, 199, ...).

%F Triangle read by rows, truncated columns of an array formed by taking sets of P(n) = Pascal's triangle, with the 1's column shifted up n = 1,2,3,... times. Then n-th row of the array = lim_{k->infinity} (P(n))^k, deleting the first 1.

%e First few rows of the array:

%e   1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...

%e   1, 1, 2,  4,  9,  23,  65,  199,   654, ...

%e   1, 1, 1,  2,  4,   8,  17,   40,   104, ...

%e   1, 1, 1,  1,  2,   4,   8,   16,    33, ...

%e   1, 1, 1,  1,  1,   2,   4,    8,    16, ...

%e   ...

%e Rightmost diagonal of 1's becomes leftmost column of the triangle:

%e   1;

%e   1, 2;

%e   1, 2, 5;

%e   1, 2, 4, 15;

%e   1, 2, 4,  9, 52;

%e   1, 2, 4,  8, 23, 203;

%e   1, 2, 4,  8, 17,  65, 877;

%e   1, 2, 4,  8, 16,  40, 199, 4140;

%e   1, 2, 4,  8, 16,  33, 104,  654, 21147;

%e   1, 2, 4,  8, 16,  32,  73,  291,  2296, 115975;

%e   1, 2, 4,  8, 16,  32,  65,  177,   857,   8569, 678570;

%e   ...

%e Example: n-th row corresponds to P(n) = Pascal's triangle with 1's column shifted up 1 row, so that P(1) =

%e   1;

%e   1;

%e   1, 1;

%e   1, 2, 1;

%e   1, 3, 3, 1;

%e   ...

%e then take lim_{k->infinity} (P(1))^k, getting A000110: (1, 1, 2, 5, 15, 52, ...), then delete the first 1.

%o (Sage)

%o # generates the diagonals of the triangle, starting with diag = 1 the Bell numbers.

%o def A171840_generator(len, diag) :

%o     A = [1]*diag

%o     for n in (0..len) :

%o         for k in range(n, 0, -1) :

%o             A[k - 1] += A[k]

%o         A.append(A[0])

%o         yield A[0]

%o for diag in (1..5) : print(list(A171840_generator(10, diag)))

%o # _Peter Luschny_, Feb 27 2012

%Y Cf. A007318, A007476, A171841, A000110.

%K nonn,tabl

%O 1,3

%A _Gary W. Adamson_, Dec 19 2009