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 A097364 Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k. 14
 1, 2, 0, 2, 1, 0, 3, 1, 1, 0, 2, 3, 1, 1, 0, 4, 2, 3, 1, 1, 0, 2, 5, 3, 3, 1, 1, 0, 4, 4, 6, 3, 3, 1, 1, 0, 3, 6, 6, 7, 3, 3, 1, 1, 0, 4, 6, 10, 7, 7, 3, 3, 1, 1, 0, 2, 9, 10, 12, 8, 7, 3, 3, 1, 1, 0, 6, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 0, 2, 11, 15, 20, 16, 14, 8, 7, 3, 3, 1, 1, 0, 4, 10, 21, 22, 24, 17 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Sum_{k=0..n-1} T(n,k) = A000041(n); T(n,0) + T(n,1) = n for n > 1; T(n,0) = A000005(n); T(n,1) = A049820(n) for n > 1; T(n,2) = floor((n-2)/2))*(floor((n-2)/2)) + 1)/2 = A000217(floor((n-2)/2))) = A008805(n-4) for n > 3. Without the 0's (which are of no consequence for the triangle) this sequence is A116685. - Emeric Deutsch, Feb 23 2006 LINKS Reinhard Zumkeller, Rows n = 1..60 of triangle, flattened G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014. FORMULA G.f.: Sum_{i>=1} x^i/((1 - x^i)*Product_{j=1..i-1} (1 - t*x^j)). - Emeric Deutsch, Feb 23 2006 EXAMPLE Triangle starts: 01:  1 02:  2  0 03:  2  1  0 04:  3  1  1  0 05:  2  3  1  1  0 06:  4  2  3  1  1  0 07:  2  5  3  3  1  1 0 08:  4  4  6  3  3  1 1 0 09:  3  6  6  7  3  3 1 1 0 10:  4  6 10  7  7  3 3 1 1 0 11:  2  9 10 12  8  7 3 3 1 1 0 12:  6  6 15 14 13  8 7 3 3 1 1 0 13:  2 11 15 20 16 14 8 7 3 3 1 1 0 14:  4 10 21 22 24 17 ... - Joerg Arndt, Feb 22 2014 T(8,0)=4: 8=4+4=2+2+2+2=1+1+1+1+1+1+1+1, T(8,1)=4: 3+3+2=2+2+2+1+1=2+2+1+1+1+1=2+1+1+1+1+1+1, T(8,2)=6: 5+3=4+2+2=3+3+1+1=3+2+2+1=3+2+1+1+1=3+1+1+1+1+1, T(8,3)=3: 4+3+1=4+2+1+1=4+1+1+1+1, T(8,4)=3: 6+2=5+2+1=5+1+1+1, T(8,5)=1: 6+1+1, T(8,6)=1: 7+1, T(8,7)=0; Sum_{k=0..7} T(8,k) = 4+4+6+3+3+1+1+0 = 22 = A000041(8). MAPLE g:=sum(x^i/(1-x^i)/product(1-t*x^j, j=1..i-1), i=1..50): gser:=simplify(series(g, x=0, 18)): for n from 1 to 15 do P[n]:=coeff(gser, x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n], t, j), j=0..n-1) od; # yields sequence in triangular form # Emeric Deutsch, Feb 23 2006 MATHEMATICA rows = 14; max = rows+2; col[k0_ /; k0 > 0] := col[k0] = Sum[x^(2*k + k0) / Product[(1-x^(k+j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x]&; col[0] := Table[Switch[n, 1, 0, 2, 1, _, n - 1 - col[1][[n]]], {n, 1, Length[col[1]]}]; Table[col[k][[n+2]], {n, 0, rows-1 }, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 10 2017, after Alois P. Heinz *) PROG (Haskell) a097364 n k = length [qs | qs <- pss !! n, last qs - head qs == k] where    pss = [] : map parts [1..] where          parts x = [x] : [i : ps | i <- [1..x],                                    ps <- pss !! (x - i), i <= head ps] a097364_row n = map (a097364 n) [0..n-1] a097364_tabl = map a097364_row [1..] -- Reinhard Zumkeller, Feb 01 2013 CROSSREFS Cf. A116685 (same sequence with zeros omitted). Columns k=3..10 give A128508, A218567, A218568, A218569, A218570, A218571, A218572, A218573. T(2*n,n) = A117989(n). - Alois P. Heinz, Nov 02 2012 Sequence in context: A195050 A127371 A036849 * A254204 A321361 A319517 Adjacent sequences:  A097361 A097362 A097363 * A097365 A097366 A097367 KEYWORD nonn,tabl AUTHOR Reinhard Zumkeller, Aug 09 2004 STATUS approved

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Last modified January 22 00:08 EST 2022. Contains 350481 sequences. (Running on oeis4.)