A178831 Rectangular array T(n,k) = binomial(n+1,2)*(n^k - (n-1)^k) read by antidiagonals.

1, 1, 3, 1, 9, 6, 1, 21, 30, 10, 1, 45, 114, 70, 15, 1, 93, 390, 370, 135, 21, 1, 189, 1266, 1750, 915, 231, 28, 1, 381, 3990, 7810, 5535, 1911, 364, 36, 1, 765, 12354, 33670, 31515, 14091, 3556, 540, 45, 1, 1533, 37830, 141970, 172935, 97671, 30940, 6084, 765, 55

Offset

1,3

Comments

T(n,k) is the sum of the elements in the image sets of all functions f:{1,2,...,k}->{1,2,...,n}.

Equivalently, the sum of the distinct entries in each length k sequence on {1,2,...,n}.

Links

G. C. Greubel, Antidiagonals n = 1..100, flattened

Formula

E.g.f. for row n: binomial(n+1,2)*exp((n-1)*x)*(exp(x) - 1).

Example

Array begins
   1,   1,    1,     1,     1,      1, ...
   3,   9,   21,    45,    93,    189, ...
   6,  30,  114,   390,  1266,   3990, ...
  10,  70,  370,  1750,  7810,  33670, ...
  15, 135,  915,  5535, 31515, 172935, ...
  21, 231, 1911, 14091, 97671, 651651, ...

Mathematica

Table[Range[7]! Rest[CoefficientList[Series[Binomial[n+1, 2] Exp[(n-1)x](Exp[x]-1), {x, 0, 7}], x]], {n, 1, 7}]//Grid
T[n_, k_]:= Binomial[n+2, 2]*((n+1)^k -n^k); Table[T[k, n-k], {n, 1, 10}, {k, 0, n-1}] (* G. C. Greubel, Jan 22 2019 *)

Prog

(PARI) {T(n, k) = binomial(n+2, 2)*((n+1)^k -(n)^k)};
for(n=1, 10, for(k=0, n-1, print1(T(k, n-k), ", "))) \\ G. C. Greubel, Jan 22 2019
(Magma) [[Binomial(k+2, 2)*((k+1)^(n-k) -k^(n-k)): k in [0..n-1]]: n in [1..10]]; // G. C. Greubel, Jan 22 2019
(Sage) [[binomial(k+2, 2)*((k+1)^(n-k) -k^(n-k)) for k in (0..n-1)] for n in (1..10)] # G. C. Greubel, Jan 22 2019
(GAP) T:=Flat(List([1..10], n->List([0..n-1], k-> Binomial(k+2, 2)*( (k+1)^(n-k) -k^(n-k)) ))); # G. C. Greubel, Jan 22 2019

Crossrefs

Cf. A068156 the case for n=2.

Sequence in context: A264364 A330509 A105545 * A027465 A164942 A236420

Adjacent sequences: A178828 A178829 A178830 * A178832 A178833 A178834

Keyword

nonn,tabl,easy

Author

Geoffrey Critzer, Dec 27 2010

Extensions

Terms a(29) onward added by G. C. Greubel, Jan 22 2019

Status

approved