cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 09 2004

Keywords

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

Examples

			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).
		

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

Programs

  • 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
  • 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 *)

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

A116685 Triangle read by rows: T(n,k) is number of partitions of n that have k parts smaller than the largest part (n>=1, k>=0).

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 2, 3, 1, 1, 2, 5, 3, 3, 1, 1, 4, 4, 6, 3, 3, 1, 1, 3, 6, 6, 7, 3, 3, 1, 1, 4, 6, 10, 7, 7, 3, 3, 1, 1, 2, 9, 10, 12, 8, 7, 3, 3, 1, 1, 6, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 2, 11, 15, 20, 16, 14, 8, 7, 3, 3, 1, 1, 4, 10, 21, 22, 24, 17, 14, 8, 7, 3, 3, 1, 1, 4, 11, 21
Offset: 1

Views

Author

Emeric Deutsch, Feb 23 2006

Keywords

Comments

Same as A097364 without the 0's.
Also number of partitions of n such that the difference between the largest and smallest parts is k (see A097364). Example: T(6,2)=3 because we have [4,2],[3,2,1] and [3,1,1,1].
Row 1 has one term; row n (n>=2) has n-1 terms.
Row sums yield the partition numbers (A000041).
T(n,0)=A000005(n) (number of divisors of n).
T(n,1)=A049820(n) (n minus number of divisors of n).
T(n,2)=A008805(n-4) for n>=4.
Sum(k*T(n,k),k=0..n-2)=A116686

Examples

			Triangle starts:
01:  1
02:  2
03:  2  1
04:  3  1  1
05:  2  3  1  1
06:  4  2  3  1  1
07:  2  5  3  3  1  1
08:  4  4  6  3  3  1 1
09:  3  6  6  7  3  3 1 1
10:  4  6 10  7  7  3 3 1 1
11:  2  9 10 12  8  7 3 3 1 1
12:  6  6 15 14 13  8 7 3 3 1 1
13:  2 11 15 20 16 14 8 7 3 3 1 1
14:  4 10 21 22 24 17 ...
T(6,2)=3 because we have [4,1,1],[3,2,1] and [2,2,1,1].
		

Crossrefs

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

Programs

Formula

G.f.: sum(i>=1, x^i/(1-x^i)/prod(j=1..i-1, 1-t*x^j) ).
Showing 1-2 of 2 results.