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 A276433 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0). 6
 1, 1, 1, 1, 3, 4, 1, 6, 1, 8, 3, 12, 3, 18, 3, 1, 24, 6, 32, 10, 45, 10, 1, 59, 17, 1, 79, 21, 1, 104, 28, 3, 137, 37, 2, 177, 50, 4, 229, 64, 4, 295, 82, 8, 377, 105, 8, 477, 139, 10, 1, 605, 174, 13, 761, 220, 21, 956, 275, 24, 1193, 350, 31, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Sum of entries in row n is A000041(n) (the partition numbers). T(n,0) = A277099(n). Sum(k*T(n,k), k>=0) = A276434(n). LINKS Alois P. Heinz, Rows n = 0..1000, flattened FORMULA G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^i)). EXAMPLE The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3. T(5,1) = 1, counting [1,1,3]. T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2]. T(8,2) = 1, counting [1,1,2,2,2]. Triangle starts: 1; 1; 1,1; 3; 4,1; 6,1; 8,3. MAPLE G := mul((t-1)*x^(i*(i+1))+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form # second Maple program: b:= proc(n, i) option remember; expand( `if`(n=0, 1, `if`(i<1, 0, add( `if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i)))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2)): seq(T(n), n=0..30); # Alois P. Heinz, Sep 30 2016 MATHEMATICA b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i + 1 == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *) CROSSREFS Cf. A000041, A276427, A276428, A276429, A276434, A277099, A277100, A277101, A277102. Sequence in context: A354718 A339964 A342915 * A343226 A030707 A349629 Adjacent sequences: A276430 A276431 A276432 * A276434 A276435 A276436 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Sep 30 2016 STATUS approved

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Last modified March 31 08:15 EDT 2023. Contains 361645 sequences. (Running on oeis4.)