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 A171840 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->inf.}, k=1,2,3,...; (P(n))^k, deleting the first 1. 1
 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, 4, 8, 16, 40, 199, 4140, 1, 2, 4, 8, 16, 33, 104, 654, 21147, 1, 2, 4, 8, 16, 32, 73, 291, 2296, 115975, 1, 2, 4, 8, 16, 32, 65, 177, 857, 8569, 678570 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row sums = A171841: (1, 3, 8, 22, 68, 241, 974,...) Right border = the Bell sequence A000110 starting (1, 2, 5, 15, 52,...). Row 2 of the array = A007476 starting (1, 1, 2, 4, 9, 23, 65, 199,...). LINKS FORMULA 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->inf.} (P(n))^k, deleting the first 1. EXAMPLE First few rows of the array = . 1, 2, 5, 15, 52, 203, 877, 4140, 21147,... 1, 1, 2, .4, .9, .23, .65, .199, ..654,... 1, 1, 1, .2, .4, ..8, .17, ..40, ..104,... 1, 1, 1, .1, .2, ..4, ..8, ..16, ...33,... 1, 1, 1, .1, .1, ..2, ..4, ...8, ...16,... ... Rightmost diagonal of 1's becomes leftmost column of the triangle: . 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, 4, 8, 16, 40, 199, 4140; 1, 2, 4, 8, 16, 33, 104, 654, 21147; 1, 2, 4, 8, 16, 32, 73, 291, 2296, 115975; 1, 2, 4, 8, 16, 32, 65, 177, 857, 8569, 678570; ... Example: n-th row corresponds to P(n) = Pascal's triangle with 1's column shifted up 1 row, so that P(1) = 1; 1; 1, 1; 1, 2, 1; 1, 3, 3, 1; ...then take Lim_{k=1..inf.} (P(1))^k, getting A000110: (1, 1, 2, 5, 15, 52,...), then delete the first 1. PROG (Sage) # generates the diagonals of the triangle, starting with diag = 1 the Bell numbers. def A171840_generator(len, diag) :     A = [1]*diag     for n in (0..len) :         for k in range(n, 0, -1) :             A[k - 1] += A[k]         A.append(A[0])         yield A[0] for diag in (1..5) : print list(A171840_generator(10, diag)) # Peter Luschny, Feb 27 2012 CROSSREFS Cf. A007476, A171841, A000110 Sequence in context: A002211 A175011 A211700 * A132309 A144224 A122881 Adjacent sequences:  A171837 A171838 A171839 * A171841 A171842 A171843 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Dec 19 2009 STATUS approved

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Last modified March 20 15:43 EDT 2019. Contains 321345 sequences. (Running on oeis4.)