STATUS
proposed
approved
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Revision History for A275416
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
Triangle read by rows: T(n,k) is the number of multisets of k odd numbers with a cap of the total sum set to n.
(history; published version)
(history; published version)
STATUS
editing
proposed
FORMULA
T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1 < k <= n.
EXAMPLE
T(6,2) = 3+2+3 = 8 counts {1,1} {1,3}, and {3,3} from taking two odd numbers <= 3; it counts {1,1} and {1,3} from taking an odd number <= 2 and an odd number <= 4; and it counts {1,1}, {1,3} and {1,5} from taking an odd number <= 1 and an odd number <= 5.
T(6,3) = 1+2+2 = 5 counts {1,1,1} from taking three odd numbers <= 2; it counts {1,1,1} and {1,1,3} from taking an odd number <= 1 and an odd number <= 2 and an odd number <= 3; and it counts {1,1,1} and {1,1,3} from taking two odd numbers <= 1 and an odd number <= 4.
STATUS
proposed
editing
STATUS
editing
proposed
NAME
Triangle read by rows: T(n,k) = is the number of multisets of k odd numbers with a cap of the total sum set to n.
EXAMPLE
T(6,3)= 1+2+2 = 5 counts {1,1,1} from taking three odd numbers <=2; it counts {1,1,1} and {1,1,3} from taking an odd number <=1 and an odd number <=2 and an odd number <=3; and it counts {1,1,1} and {1,1,3} rom from taking two odd numbers <=1 and an odd number <= 4.
STATUS
approved
editing
STATUS
reviewed
approved
STATUS
proposed
reviewed
STATUS
editing
proposed
MATHEMATICA
b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[Ceiling[i/2] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)
STATUS
approved
editing
STATUS
editing
approved