Search: seq:1,1,1,1,2,1,1,5,3,1
|
| |
|
|
A047884
|
|
Triangle of numbers a(n,k) = number of Young tableaux with n cells and k rows (1 <= k <= n); also number of self-inverse permutations on n letters in which the length of the longest scattered (i.e., not necessarily contiguous) increasing subsequence is k.
|
|
+30
19
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 9, 11, 4, 1, 1, 19, 31, 19, 5, 1, 1, 34, 92, 69, 29, 6, 1, 1, 69, 253, 265, 127, 41, 7, 1, 1, 125, 709, 929, 583, 209, 55, 8, 1, 1, 251, 1936, 3356, 2446, 1106, 319, 71, 9, 1, 1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
REFERENCES
|
W. Fulton, Young Tableaux, Cambridge, 1997.
D. Stanton and D. White, Constructive Combinatorics, Springer, 1986.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
For n=3 the 4 tableaux are
1 2 3 . 1 2 . 1 3 . 1
. . . . 3 . . 2 . . 2
. . . . . . . . . . 3
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 5, 3, 1;
1, 9, 11, 4, 1;
1, 19, 31, 19, 5, 1;
1, 34, 92, 69, 29, 6, 1;
1, 69, 253, 265, 127, 41, 7, 1;
1, 125, 709, 929, 583, 209, 55, 8, 1;
1, 251, 1936, 3356, 2446, 1106, 319, 71, 9, 1;
1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10, 1;
...
|
|
|
MAPLE
|
h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l) `if`(n=0 or i=1, (p->h(p)*x^`if`(p=[], 0, p[1]))
([l[], 1$n]), add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2, [])):
seq(T(n), n=1..14); # Alois P. Heinz, Apr 16 2012, revised Mar 05 2014
|
|
|
MATHEMATICA
|
Table[ Plus@@( NumberOfTableaux/@ Reverse/@Union[ Sort/@(Compositions[ n-m, m ]+1) ]), {n, 12}, {m, n} ]
(* Second program: *)
h[l_] := With[{n=Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[ l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n== 0|| i==1, Function[p, h[p]*x^If[p == {}, 0, p[[1]] ] ] [ Join[l, Array[1&, n]]], Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n, {}]];
|
|
|
CROSSREFS
|
Columns k=1-10 give: A000012, A014495, A217323, A217324, A217325, A217326, A217327, A217328, A217321, A217322. - Alois P. Heinz, Oct 03 2012
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A078920
|
|
Upper triangle of Catalan Number Wall.
|
|
+30
14
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 14, 4, 1, 1, 42, 84, 30, 5, 1, 1, 132, 594, 330, 55, 6, 1, 1, 429, 4719, 4719, 1001, 91, 7, 1, 1, 1430, 40898, 81796, 26026, 2548, 140, 8, 1, 1, 4862, 379236, 1643356, 884884, 111384, 5712, 204, 9, 1, 1, 16796, 3711916, 37119160, 37119160, 6852768, 395352, 11628, 285, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
As square array: number of certain symmetric plane partitions, see Forrester/Gamburd paper.
Formatted as a square array, the column k gives the Hankel transform of the Catalan numbers (A000108) beginning at A000108(k); example: Hankel transform of [42, 132, 429, 1430, 4862, ...] is [42, 594, 4719, 26026, 111384, ...] (see A091962). - Philippe Deléham, Apr 12 2007
As square array T(n,k): number of all k-watermelons with a wall of length n. - Ralf Stephan, May 09 2007
Consider "Young tableaux with entries from the set {1,...,n}, strictly increasing in rows and not decreasing in columns. Note that usually the reverse convention between rows and columns is used." de Sainte-Catherine and Viennot (1986) proved that "the number b_{n,k} of such Young tableaux having only columns with an even number of elements and bounded by height p = 2*k" is given by b_{n,k} = Product_{1 <= i <= j <= n} (2*k + i + j)/(i + j)." It turns out that for the current array, T(n,k) = b(n-k,k) for n >= 0 and 0 <= k <= n. - Petros Hadjicostas, Sep 04 2019
As square array, b(k, n) = T(n+k-1, n) for k >= 1 and n >= 1 is the number of n-tuples P = (p_1, p_2, ..., p_n) of non-intersecting lattice paths that lie below the diagonal, such that each p_i starts at (i, i) and ends at (2n+k-i, 2n+k-i). (This is just a different way of looking at n-watermelons with a wall of length k since many of the steps of these paths are going to be fixed while the rest form an n-watermelon. See the Krattenthaler et al. paper.) Equivalently b(k, n) is the number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths, each with 2k steps such that for every i (1 <= i <= n-1), p_i is included in p_{i+1}. A Dyck path p is said to be included in a Dyck path q if the height of path p after j steps is at most the height of path q after j steps, for all j (1 <= j <= 2k). - Farzan Byramji, Jun 17 2021
|
|
|
LINKS
|
M. de Sainte-Catherine and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
|
|
|
FORMULA
|
T(n,k) = Product_{i=1..n-k} Product_{j=i..n-k} (i+j+2*k)/(i+j). [corrected by Petros Hadjicostas, Jul 24 2019]
T(n, k) = Product_{j=0..k-1} binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j).
T(n, k) = ((n+1)!/(n-k+1)!)*Product_{j=0..k-1} Catalan(n-j)/binomial(n+j+1, n-j). (End)
|
|
|
EXAMPLE
|
Triangle T(n,k) (with rows n >= 0 and columns k >= 0) starts as follows:
1;
1, 1;
1, 2, 1;
1, 5, 3, 1;
1, 14, 14, 4, 1;
1, 42, 84, 30, 5, 1;
1, 132, 594, 330, 55, 6, 1;
1, 429, 4719, 4719, 1001, 91, 7, 1;
1, 1430, 40898, 81796, 26026, 2548, 140, 8, 1;
1, 4862, 379236, 1643356, 884884, 111384, 5712, 204, 9, 1;
...
|
|
|
MAPLE
|
T:= (n, k)-> mul(mul((i+j+2*k)/(i+j), j=i..n-k), i=1..n-k):
|
|
|
MATHEMATICA
|
T[n_, k_] := Product[(2*i+1)!*(2*n-2*i)!/(n-i)!/(n+i+1)!, {i, 0, k-1}]; Table[T[n, k], {n, 1, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 28 2015, adapted from PARI *)
|
|
|
PROG
|
(PARI) T(n, k)=if(k<0 || k>n, 0, prod(i=0, k-1, (2*i+1)!*(2*n-2*i)!/(n-i)!/(n+i+1)!))
(PARI) {C(n)=if(n<0, 0, (2*n)!/n!/(n+1)!)}; T(n, k)=if(k<0 || k>n, 0, matdet(matrix(k, k, i, j, C(i+j-1+n-k))))
(Sage)
def A078920(n, k): return product( binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j) for j in (0..k-1) )
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A055818
|
|
Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = R(0,i) = 1 for i >= 0, R(i,j) = Sum_{h=0..i-1} Sum_{m=0..j} R(h,m) for i >= 1, j >= 1.
|
|
+30
13
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 11, 9, 4, 1, 1, 23, 24, 14, 5, 1, 1, 47, 60, 43, 20, 6, 1, 1, 95, 144, 122, 69, 27, 7, 1, 1, 191, 336, 328, 217, 103, 35, 8, 1, 1, 383, 768, 848, 640, 354, 146, 44, 9, 1, 1, 767, 1728, 2128, 1800, 1131, 543, 199, 54, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Rows begins as:
1;
1, 1;
1, 2, 1;
1, 5, 3, 1;
1, 11, 9, 4, 1;
...
|
|
|
MAPLE
|
T:= proc(i, j) option remember;
if i=0 or j=0 then 1
else add(add(T(h, m), m=0..j), h=0..i-1)
fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020
|
|
|
MATHEMATICA
|
T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h, 0, i-1}, {m, 0, j}]]; Table[T[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 21 2020 *)
|
|
|
PROG
|
(PARI) T(i, j) = if(i==0 || j==0, 1, sum(h=0, i-1, sum(m=0, j, T(h, m) )));
for(n=0, 12, for(k=0, n, print1(T(n-k, k), ", "))) \\ G. C. Greubel, Jan 21 2020
(Magma)
function T(i, j)
if i eq 0 or j eq 0 then return 1;
else return (&+[(&+[T(h, m): m in [0..j]]): h in [0..i-1]]);
end if; return T; end function;
[T(n-k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020
(Sage)
@CachedFunction
def T(i, j):
if (i==0 or j==0): return 1
else: return sum(sum(T(h, m) for m in (0..j)) for h in (0..i-1))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020
(GAP)
T:= function(i, j)
if i=0 or j=0 then return 1;
else return Sum([0..i-1], h-> Sum([0..j], m-> T(h, m) ));
fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n-k, k) ))); # G. C. Greubel, Jan 21 2020
|
|
|
CROSSREFS
|
Cf. A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055828, A055829.
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A124328
|
|
Triangle read by rows: T(n,k) is the number of ordered trees with n edges, with thinning limbs and with root of degree k (1<=k<=n). An ordered tree with thinning limbs is such that if a node has k children, all its children have at most k children.
|
|
+30
8
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 10, 9, 4, 1, 1, 22, 25, 14, 5, 1, 1, 46, 69, 44, 20, 6, 1, 1, 101, 186, 137, 70, 27, 7, 1, 1, 220, 503, 416, 235, 104, 35, 8, 1, 1, 492, 1356, 1256, 766, 375, 147, 44, 9, 1, 1, 1104, 3663, 3760, 2465, 1296, 567, 200, 54, 10, 1, 1, 2515, 9907
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
The g.f. F[k]=F[k](z) of column k satisfies F[k]={F[k-1]^(1/(k-1) + zF[k]}^k; F[1]=z/(1-z).
|
|
|
EXAMPLE
|
Triangle starts:
1;
1,1;
1,2,1;
1,5,3,1;
1,10,9,4,1;
|
|
|
MATHEMATICA
|
t[n_, n_] = 1; t[n_, k_] /; n == k + 1 := t[n, k] = n - 1; t[n_, k_] := t[n, k] = Coefficient[(1 + x*Sum[ x^(r - 1)*Sum[ t[r, c], {c, 1, k }], {r, 1, n - k}] + x^n)^k, x, n - k ]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 21 2013, after Paul D. Hanna *)
|
|
|
PROG
|
(PARI) {T(n, k)=if(n==k, 1, if(n==k+1, n-1, polcoeff( (1 + x*sum(r=1, n-k, x^(r-1)*sum(c=1, k, T(r, c)))+x*O(x^n))^k, n-k)))} - Paul D. Hanna, Nov 12 2006
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A125800
|
|
Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals.
|
|
+30
8
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 23, 12, 4, 1, 1, 239, 93, 22, 5, 1, 1, 5828, 1632, 238, 35, 6, 1, 1, 342383, 68457, 5827, 485, 51, 7, 1, 1, 50110484, 7112055, 342382, 15200, 861, 70, 8, 1, 1, 18757984046, 1879090014, 50110483, 1144664, 32856, 1393, 92, 9, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
Determinant of n X n upper left submatrix is 3^(n*(n-1)*(n-2)/6).
This table is related to partitions of numbers into powers of 3 (see A078122).
Triangle A078122 shifts left one column under matrix cube.
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = T(n,k-1) + T(n-1,3*k) for n > 0, k > 0, with T(0,n)=T(n,0)=1 for n >= 0.
G.f. of row n is g_n(z) where g_{n+1}(z) = (1-z)^(-1)*Sum_{w^3=1} g_n(w*z^(1/3)) (the sum being over the cube roots of unity). - Robert Israel, Jun 02 2019
|
|
|
EXAMPLE
|
Recurrence T(n,k) = T(n,k-1) + T(n-1,3*k) is illustrated by:
T(3,3) = T(3,2) + T(2,9) = 93 + 145 = 238;
T(4,3) = T(4,2) + T(3,9) = 1632 + 4195 = 5827;
T(5,3) = T(5,2) + T(4,9) = 68457 + 273925 = 342382.
Rows of this table begin:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...;
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, ...;
1, 23, 93, 238, 485, 861, 1393, 2108, 3033, 4195, 5621, ...;
1, 239, 1632, 5827, 15200, 32856, 62629, 109082, 177507, 273925,...;
1, 5828, 68457, 342382, 1144664, 3013980, 6769672, 13570796, ...;
1, 342383, 7112055, 50110483, 215155493, 690729981, 1828979530, ...;
1, 50110484, 1879090014, 18757984045, 103674882878, 406279238154,..;
1, 18757984046, 1287814075131, 18318289003447, 130648799730635, ...;
1;
1, 1;
1, 3, 1;
1, 12, 9, 1;
1, 93, 117, 27, 1;
1, 1632, 3033, 1080, 81, 1;
1, 68457, 177507, 86373, 9801, 243, 1; ...
where row sums form column 1 of this table A125790,
1;
2, 1;
5, 6, 1;
23, 51, 18, 1;
239, 861, 477, 54, 1;
5828, 32856, 25263, 4347, 162, 1; ...
where row sums form column 2 of this table A125790,
|
|
|
MAPLE
|
f[0]:= 1/(1-z):
S[0]:= series(f[0], z, 21):
for n from 1 to 20 do
ff:= unapply(f[n-1], z);
f[n]:= simplify(1/3*sum(ff(w*z^(1/3)), w=RootOf(Z^3-1, Z)))/(1-z);
S[n]:= series(f[n], z, 21-n)
od:
seq(seq(coeff(S[s-i], z, i), i=0..s), s=0..20); # Robert Israel, Jun 02 2019
|
|
|
MATHEMATICA
|
T[0, _] = T[_, 0] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 3 k]; Table[T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)
|
|
|
PROG
|
(PARI) T(n, k, p=0, q=3)=local(A=Mat(1), B); if(n<p || p<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return((A^(k+1))[n+1, p+1]))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A125860
|
|
Rectangular table where column k equals row sums of matrix power A097712^k, read by antidiagonals.
|
|
+30
8
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 12, 4, 1, 1, 86, 69, 22, 5, 1, 1, 698, 612, 178, 35, 6, 1, 1, 9551, 8853, 2251, 365, 51, 7, 1, 1, 226592, 217041, 46663, 5990, 651, 70, 8, 1, 1, 9471845, 9245253, 1640572, 161525, 13131, 1057, 92, 9, 1, 1, 705154187
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
Triangle A097712 satisfies: A097712(n,k) = A097712(n-1,k) + [A097712^2](n-1,k-1) for n > 0, k > 0, with A097712(n,0)=A097712(n,n)=1 for n >= 0. Column 1 equals A016121, which counts the sequences (a_1, a_2, ..., a_n) of length n with a_1 = 1 satisfying a_i <= a_{i+1} <= 2*a_i.
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = Sum_{j=0..k} T(n-1, j+k) for n > 0, with T(0,n)=T(n,0)=1 for n >= 0.
|
|
|
EXAMPLE
|
Recurrence is illustrated by:
T(4,1) = T(3,1) + T(3,2) = 17 + 69 = 86;
T(4,2) = T(3,2) + T(3,3) + T(3,4) = 69 + 178 + 365 = 612;
T(4,3) = T(3,3) + T(3,4) + T(3,5) + T(3,6) = 178 + 365 + 651 + 1057 = 2251.
Rows of this table begin:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,...;
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, ...;
1, 17, 69, 178, 365, 651, 1057, 1604, 2313, 3205, 4301, 5622, 7189,..;
1, 86, 612, 2251, 5990, 13131, 25291, 44402, 72711, 112780, 167486,..;
1, 698, 8853, 46663, 161525, 435801, 996583, 2025458, 3768273, ...;
1, 9551, 217041, 1640572, 7387640, 24530016, 66593821, 156664796, ...;
1, 226592, 9245253, 100152049, 586285040, 2394413286, 7713533212, ...;
1, 9471845, 695682342, 10794383587, 82090572095, 412135908606, ...;
1, 705154187, 93580638024, 2079805452133, 20540291522675, ...;
1, 94285792211, 22713677612832, 723492192295786, 9278896006526795,...;
1, 22807963405043, 10025101876435413, 458149292979837523, ...;
...
where column k equals the row sums of matrix power A097712^k for k >= 0.
1;
1, 1;
1, 3, 1;
1, 8, 7, 1;
1, 25, 44, 15, 1;
1, 111, 346, 208, 31, 1;
1, 809, 4045, 3720, 912, 63, 1;
1, 10360, 77351, 99776, 35136, 3840, 127, 1;
1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255; ...
1;
2, 1;
5, 6, 1;
17, 37, 14, 1;
86, 302, 193, 30, 1;
698, 3699, 3512, 881, 62, 1;
9551, 73306, 96056, 34224, 3777, 126, 1; ...
1;
3, 1;
12, 9, 1;
69, 87, 21, 1;
612, 1146, 447, 45, 1;
8853, 22944, 12753, 2019, 93, 1;
217041, 744486, 549453, 120807, 8595, 189, 1; ...
|
|
|
PROG
|
(PARI) T(n, k)=if(n==0 || k==0, 1, sum(j=0, k, T(n-1, j+k)))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A308292
|
|
A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.
|
|
+30
6
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 16, 19, 4, 1, 1, 65, 271, 69, 5, 1, 1, 326, 7365, 5248, 251, 6, 1, 1, 1957, 326011, 1107697, 110251, 923, 7, 1, 1, 13700, 21295783, 492911196, 191448941, 2435200, 3431, 8, 1, 1, 109601, 1924223799, 396643610629, 904434761801, 35899051101, 55621567, 12869, 9, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
For r > 1, row r is asymptotic to sqrt(2*Pi) * (r*n)^(r*n + 1/2) / ((r!)^n * exp(r*n-1)). - Vaclav Kotesovec, May 24 2020
|
|
|
LINKS
|
|
|
|
FORMULA
|
A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * x^i/i! = (Sum_{i=0..n} x^i/i!)^k.
|
|
|
EXAMPLE
|
For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + 2*x + 4*x^2/2 + 8*x^3/6 + 14*x^4/24 + 20*x^5/120 + 20*x^6/720. So A(3,2) = 1 + 2 + 4 + 8 + 14 + 20 + 20 = 69.
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 5, 16, 65, 326, ...
1, 3, 19, 271, 7365, 326011, ...
1, 4, 69, 5248, 1107697, 492911196, ...
1, 5, 251, 110251, 191448941, 904434761801, ...
1, 6, 923, 2435200, 35899051101, 1856296498826906, ...
1, 7, 3431, 55621567, 7101534312685, 4098746255797339511, ...
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A107735
|
|
Array read by antidiagonals: A(n,k) = Verlinde numbers for quasiparabolic bundles (n >= 3, k >= 0)
|
|
+30
5
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 4, 13, 4, 1, 1, 21, 11, 25, 5, 1, 1, 8, 141, 24, 41, 6, 1, 1, 85, 43, 521, 45, 61, 7, 1, 1, 16, 1485, 160, 1401, 76, 85, 8, 1, 1, 341, 171, 10569, 461, 3101, 119, 113, 9, 1, 1, 32, 15565, 1088, 46649, 1112, 6021, 176, 145, 10
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
3,5
|
|
|
REFERENCES
|
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
|
|
|
LINKS
|
|
|
|
FORMULA
|
The reference gives an explicit formula. For odd n this is
A(n,k) = (1/(2*k+1))*sum( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*k+2) )^(-n+2), j=0..2*k). - N. J. A. Sloane, Apr 20 2013.
For even n use the same formula but replace k by k/2. - Michel Marcus, Apr 20 2013
|
|
|
EXAMPLE
|
Array begins:
1 1 1 1 1 1 1 1 1 1 ...
1 2 3 4 5 6 7 8 9 10 ...
1 5 13 25 41 61 85 113 ...
1 4 11 24 45 76 119 ...
1 21 141 521 1401 3101 ...
|
|
|
MAPLE
|
Digits:=100;
A:=proc(n, k) local kp;
if (n mod 2) = 1 then
round( (1/(2*k+1))*add( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*k+2) )^(-n+2), j=0..2*k))
else kp:=k/2;
round( (1/(2*kp+1))*add( (-1)^(n*j)*sin( (2*j+1)*Pi/(4*kp+2) )^(-n+2), j=0..2*kp)); fi;
end;
|
|
|
MATHEMATICA
|
t[n_, k_] := With[{kp = If[!Divisible[n, 2], k, k/2]}, Round[1/(2*kp+1)*Sum[(-1)^(n*j)*Sin[(2*j+1)*Pi/(4*kp+2)]^(-n+2), {j, 0, 2*kp}]]]; Table[t[n-k, k], {n, 3, 13}, {k, 0, n-3}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Michel Marcus *)
|
|
|
PROG
|
(PARI) t(n, k) = {if (! (n % 2), k = k/2); return (round((1/(2*k+1))*sum(j=0, 2*k, (-1)^(n*j)*sin((2*j+1)*Pi/(4*k+2))^(-n+2)))); } \\ Michel Marcus, Apr 20 2013
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A117396
|
|
Triangle, read by rows, defined by: T(n,k) = (k+1)*T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)*T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.
|
|
+30
4
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 11, 4, 1, 1, 77, 51, 19, 5, 1, 1, 437, 291, 109, 29, 6, 1, 1, 2957, 1971, 739, 197, 41, 7, 1, 1, 23117, 15411, 5779, 1541, 321, 55, 8, 1, 1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1, 1, 2018957, 1345971, 504739, 134597, 28041, 4807, 701, 89, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
Columns equal the partial sums of columns of triangle A092582 for k>0: T(n, k) - T(n-1, k) = A092582(n,k) = number of permutations p of [n] having length of first run equal to k.
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = k*Sum_{j=k-1..n} j!/(k+1)! for n >= k > 0, with T(n,0) = 1 for n >= 0. - Paul D. Hanna, Jun 20 2006
|
|
|
EXAMPLE
|
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 5, 3, 1;
1, 17, 11, 4, 1;
1, 77, 51, 19, 5, 1;
1, 437, 291, 109, 29, 6, 1;
1, 2957, 1971, 739, 197, 41, 7, 1;
1, 23117, 15411, 5779, 1541, 321, 55, 8, 1;
1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1; ...
Matrix inverse is:
1;
-1, 1;
1, -2, 1;
1, 1, -3, 1;
1, 1, 1, -4, 1;
1, 1, 1, 1, -5, 1; ...
Matrix log is the integer triangle A117398:
0;
1, 0;
0, 2, 0;
-1, 2, 3, 0;
-3, 4, 5, 4, 0;
-9, 14, 15, 9, 5, 0;
-33, 68, 65, 34, 14, 6, 0; ...
|
|
|
MATHEMATICA
|
T[n_, k_]:= T[n, k]= If[k==0, 1, k*Sum[j!/(k+1)!, {j, k-1, n}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 24 2021 *)
|
|
|
PROG
|
(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, (k+1)*T(n, k+1)-sum(j=1, n-k-1, T(j, 0)*T(n, j+k+1))))
(PARI) /* Definition by Matrix Inverse: * / T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, if(r==c+1, -c, 1)))); (M^-1)[n+1, k+1]
(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, k*sum(j=k-1, n, j!)/(k+1)!)) \\ Paul D. Hanna, Jun 20 2006
(Magma) [k eq 0 select 1 else k*(&+[Factorial(j)/Factorial(k+1): j in [k-1..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 24 2021
(Sage)
def A117396(n, k): return 1 if (k==0) else k*sum(factorial(j)/factorial(k+1) for j in (k-1..n))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A241579
|
|
Square array read by antidiagonals downwards: T(n,k) = Sum_{j=1..k} n^(k-j)*Stirling_2(k,j) (n >= 0, k >= 1).
|
|
+30
4
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 15, 11, 4, 1, 1, 52, 49, 19, 5, 1, 1, 203, 257, 109, 29, 6, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ...
1, 3, 11, 49, 257, 1539, 10299, 75905, 609441, 5284451, 49134923, 487026929, ...
1, 4, 19, 109, 742, 5815, 51193, 498118, 5296321, 60987817, 754940848, 9983845261, ...
1, 5, 29, 201, 1657, 15821, 170389, 2032785, 26546673, 376085653, 5736591885, 93614616409, ...
1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, ...
1, 7, 55, 505, 5497, 69823, 1007407, 16157905, 284214097, 5432922775, 112034017735, 2476196276617, ...
1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, ...
...
|
|
|
MAPLE
|
with(combinat):
T:=(n, k)->add(n^(k-j)*stirling2(k, j), j=1..k);
r:=n->[seq(T(n, k), k=1..12)];
for n from 0 to 8 do lprint(r(n)); od:
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.010 seconds
|
