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A239738 Triangle read by rows: T(n,k) is the number of n-tuples with sum k + n whose i-th element is a positive integer <= prime(i), 0 <= k < A070826(n).

%I A239738 #59 Nov 08 2024 07:23:59
%S A239738 1,1,2,1,3,5,6,1,4,9,15,21,26,29,1,5,14,29,50,76,105,134,160,181,196,
%T A239738 204,1,6,20,49,99,175,280,414,574,755,951,1155,1359,1554,1730,1876,
%U A239738 1981,2036,1,7,27,76,175,350,630,1044,1618,2373,3324,4479,5838,7392,9122,10998,12979,15014,17044,19005,20832,22463,23842,24921,25662,26039
%N A239738 Triangle read by rows: T(n,k) is the number of n-tuples with sum k + n whose i-th element is a positive integer <= prime(i), 0 <= k < A070826(n).
%C A239738 Original name: Normal distributions from the primes. The contracted sequence of integers generated from the frequencies of the summation of elements of the subsets of the Cartesian product of the natural numbers of ascending prime cardinality. That is, given a number of sets of the natural numbers of ascending modulo P(n+1), the probabilities of generating a given number from the selection of one element from each set form the given sequence.
%C A239738 Although this sequence initially appears similar to A131791, its derivation is entirely different and it deviates quickly.
%C A239738 By sets of natural numbers of ascending prime cardinality, it is meant
%C A239738 N_1 = {1,2}, N_2 = {1,2,3}, N_3 = {1,2,3,4,5}, N_4 = {1,2,3,4,5,6,7}, ..., N_w = {1,2,3,...,p_w}, where the p_i are primes.
%C A239738 with Cartesian products
%C A239738 N_1 X N_2 = {1,2} X {1,2,3} = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)}, etc.
%C A239738 and the sum of the elements of the product's subsets denoted
%C A239738 Sum[N_1 X N_2] = {(2),(3),(4),(3),(4),(5)}
%C A239738 whose elements have the frequencies [1,2,2,1] (with respect to magnitude). It is these frequencies that form the sequence, the symmetry allows for the omission of repeated terms, and hence the following contraction halves the data without loss of information:
%C A239738   [1,1] -> [1]
%C A239738   [1,2,2,1] -> [1,2]
%C A239738   [1,3,5,6,6,5,3,1] -> [1,3,5,6]
%C A239738 and so forth.
%C A239738 When arranged by row of the number of sets used, then P(S(u,r) = u + r - 1) = T(r,u)/prime(r)#, P(S = X) the probability that the sum S equals the value X, and prime(r)# is the product of the first r primes (A002110), then the structure and symmetry become more apparent.
%C A239738 Each row contains l(r) = (1/2)*(Sum p(r) - r + 1) terms and clearly the sum of each row must equal half the product of the primes used,
%C A239738   Sum_{u=1..l(r)} T(r,u) = (1/2)*prime(r)#,
%C A239738 and one can see that in general for all u, r:
%C A239738   P(S(u,r) = r) = P(S(u,r) = Sum p(r)) = 1/Product p(r),
%C A239738   P(S(u,r) = r + 1) = P(S(u,r) = Sum p(r) - 1) = r/prime(r)#,
%C A239738   P(S = r + i) = P(S = Sum p(r) - i) = T(r,u+i)/prime(r)#, [0 <= i <= l(r) - 1)],
%C A239738   S(u,r) ~ N(mu(r),sigma(r)^2).
%H A239738 Andrew Howroyd, <a href="/A239738/b239738.txt">Table of n, a(n) for n = 1..2005</a> (rows 1..20)
%H A239738 Mikhail Gaichenkov, <a href="https://math.stackexchange.com/questions/4628414/normal-distributions-from-the-primes-and-its-generation-function/4631059?noredirect=1#comment9775348_4631059">Normal distributions from the primes and its generation function</a>, Mathematics StackExchange, 2023.
%F A239738 T(n,k) = [x^k] (1/(x-1)^n) * Product_{i=1..n} (x^prime(i)-1). - _Steven Foster Clark_, Feb 05 2023
%F A239738 T(n,k) = [x^k] Product_{i=1..n} Sum_{j=0..prime(i)-1} x^j. - _Andrew Howroyd_, Feb 05 2023
%e A239738 Triangle T(n,k) begins: (n >= 1, k >= 0)
%e A239738   1;
%e A239738   1, 2;
%e A239738   1, 3,  5,  6;
%e A239738   1, 4,  9, 15, 21,  26,  29;
%e A239738   1, 5, 14, 29, 50,  76, 105, 134, 160, 181, 196,  204;
%e A239738   1, 6, 20, 49, 99, 175, 280, 414, 574, 755, 951, 1155, 1359, 1554, 1730, 1876, 1981, 2036;
%e A239738   ...
%e A239738 T(3, 2) = 5 because the following 3-tuples have sum 2 + 3 = 5: (1,1,3), (1,2,2), (1,3,1), (2,1,2), (2,2,1). The tuple (3,1,1) is excluded because the 1st term is required to be no greater than prime(1) = 2.
%t A239738 row[r_]:=Drop[#,-Length[#]/2]&[Transpose[Tally[Total[Tuples[Table[Range[1,Prime[k]],{k,1,r}]],{2}]]][[2]]] (* generates row r of the table *)
%t A239738 Grid@Table[row[r],{r,1,7}] (* generates the table *)
%t A239738 Flatten@Table[row[r],{r,1,7}] (* generates the sequence *) (* _Steven Foster Clark_, Feb 02 2023 *)
%t A239738 row[r_]:=Drop[#,-Length[#]/2]&[CoefficientList[1/(x-1)^r Product[(x^Prime[i]-1),{i,1,r}],x]] (* generates row r of the table *) (* _Steven Foster Clark_, Feb 07 2023 *)
%o A239738 (PARI) row(n)={my(v=Vecrev(prod(i=1, n, 1 - x^prime(i))/(1 - x)^n)); v[1..#v/2]} \\ _Andrew Howroyd_, Feb 06 2023
%Y A239738 Row sums are A070826.
%Y A239738 Cf. A005521 (row lengths).
%K A239738 nonn,tabf
%O A239738 1,3
%A A239738 _Stuart Cooper_, Mar 26 2014
%E A239738 Name edited by _Andrew Howroyd_, Feb 05 2023