Search: a124327 -id:a124327
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A367955
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Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows.
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+10
11
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 2, 3, 1, 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1, 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1, 1, 1, 2, 5, 10, 23, 47, 39, 49, 81, 84, 129, 74, 78, 70, 87, 33, 23, 29, 5, 6, 1, 1, 1, 2, 5, 10, 23, 47, 103, 81, 129, 172, 261, 304, 431, 299, 325, 376, 317, 424, 196, 183, 144, 165, 52, 34, 41, 6, 7, 1
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OFFSET
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0,7
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COMMENTS
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Rows and also columns reversed converge to A365441.
T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.
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LINKS
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FORMULA
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Sum_{k=n..n*(n+1)/2} k * T(n,k) = A278677(n+1) for n>=1.
Sum_{k=n..n*(n+1)/2} (k-n) * T(n,k) = A200660(n) for n>=1.
T(n,n) = T(n,n*(n+1)/2) = 1.
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EXAMPLE
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T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
T(5,15) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
1;
. 1;
. . 1, 1;
. . . 1, 1, 2, 1;
. . . . 1, 1, 2, 5, 2, 3, 1;
. . . . . 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1;
. . . . . . 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
...
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MAPLE
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b:= proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
(t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
end:
T:= (n, k)-> b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
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CROSSREFS
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Row n has A000124(n-1) terms (for n>=1).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A124325
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Number of blocks of size >1 in all partitions of an n-set.
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+10
6
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0, 0, 1, 4, 17, 76, 362, 1842, 9991, 57568, 351125, 2259302, 15288000, 108478124, 805037105, 6233693772, 50257390937, 421049519856, 3659097742426, 32931956713294, 306490813820239, 2945638599347760, 29198154161188501
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OFFSET
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0,4
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COMMENTS
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Sum of the first entries in all blocks of all set partitions of [n-1]. a(4) = 17 because the sum of the first entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+4+3+3+6 = 17. - Alois P. Heinz, Apr 24 2017
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LINKS
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FORMULA
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a(n) = B(n+1)-B(n)-n*B(n-1), where B(q) are the Bell numbers (A000110).
E.g.f.: (exp(z)-1-z)*exp(exp(z)-1).
a(n) = Sum_{k=0..floor(n/2)} k*A124324(n,k).
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EXAMPLE
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a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.
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MAPLE
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with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n), n=0..23);
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MATHEMATICA
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nn=22; Range[0, nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1], {x, 0, nn}], x] (* Geoffrey Critzer, Mar 28 2013 *)
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PROG
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(PARI)
N = 66; x = 'x + O('x^N);
egf = (exp(x)-1-x)*exp(exp(x)-1) + 'c0;
gf = serlaplace(egf);
v = Vec(gf); v[1]-='c0; v
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A368204
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Number of partitions of [n] whose block minima sum to n.
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+10
4
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1, 1, 0, 2, 2, 2, 29, 56, 191, 380, 5097, 14288, 74359, 283884, 1106529, 13588409, 53640963, 350573155, 1867738775, 10770352150, 50050737949, 744605446778, 3615378756421, 29368052533243, 195027586980839, 1442227919200245, 8964685271444243, 61478734886319324
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OFFSET
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0,4
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LINKS
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FORMULA
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EXAMPLE
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a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 0.
a(3) = 2: 13|2, 1|23.
a(4) = 2: 124|3, 12|34.
a(5) = 2: 1235|4, 123|45.
a(6) = 29: 12346|5, 1234|56, 1456|2|3, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 14|25|36, 146|2|35, 14|26|35, 14|2|356, 156|24|3, 15|246|3, 15|24|36, 16|245|3, 1|2456|3, 1|245|36, 16|24|35, 1|246|35, 1|24|356, 156|2|34, 15|26|34, 15|2|346, 16|25|34, 1|256|34, 1|25|346, 16|2|345, 1|26|345, 1|2|3456.
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MAPLE
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b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
`if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
end:
a:= n-> b(n, 1, 0, n):
seq(a(n), n=0..42);
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MATHEMATICA
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b[n_, i_, t_, m_] := b[n, i, t, m] = If[n == 0, t^(m - i + 1),
If[(i + m)*(m + 1 - i)/2 < n || i > n, 0, If[t == 0, 0,
t*b[n, i + 1, t, m]] + b[n - i, i + 1, t + 1, m]]];
a[n_] := If[n == 0, 1, b[n, 1, 0, n]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A369596
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Number T(n,k) of permutations of [n] whose fixed points sum to k; triangle T(n,k), n>=0, 0<=k<=A000217(n), read by rows.
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+10
4
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1, 0, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 9, 2, 2, 3, 3, 2, 1, 1, 0, 0, 1, 44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1, 265, 44, 44, 53, 53, 62, 64, 29, 22, 24, 16, 16, 8, 6, 5, 4, 2, 1, 1, 0, 0, 1, 1854, 265, 265, 309, 309, 353, 362, 406, 150, 159, 126, 126, 93, 86, 44, 36, 29, 19, 19, 9, 7, 5, 4, 2, 1, 1, 0, 0, 1
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OFFSET
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0,8
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LINKS
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FORMULA
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EXAMPLE
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T(3,0) = 2: 231, 312.
T(3,1) = 1: 132.
T(3,2) = 1: 321.
T(3,3) = 1: 213.
T(3,6) = 1: 123.
T(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Triangle T(n,k) begins:
1;
0, 1;
1, 0, 0, 1;
2, 1, 1, 1, 0, 0, 1;
9, 2, 2, 3, 3, 2, 1, 1, 0, 0, 1;
44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1;
...
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MAPLE
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b:= proc(s) option remember; (n-> `if`(n=0, 1, add(expand(
`if`(j=n, x^j, 1)*b(s minus {j})), j=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
seq(T(n), n=0..7);
# second Maple program:
g:= proc(n) option remember; `if`(n=0, 1, n*g(n-1)+(-1)^n) end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, g(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m-1)))
end:
T:= (n, k)-> b(k, min(n, k), n):
seq(seq(T(n, k), k=0..n*(n+1)/2), n=0..7);
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MATHEMATICA
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g[n_] := g[n] = If[n == 0, 1, n*g[n - 1] + (-1)^n];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0,
If[n == 0, g[m], b[n, i-1, m] + b[n-i, Min[n-i, i-1], m-1]]];
T[n_, k_] := b[k, Min[n, k], n];
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CROSSREFS
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Column k=3 gives A000255(n-2) for n>=2.
Cf. A000217, A001710, A008289, A008290, A038720, A053632, A062869, A124327, A143946, A143947, A161680, A263753, A263756, A317527, A367955, A368338.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A365821
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Total number of partitions of [n-s] whose block minima sum to s, summed over all s.
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+10
3
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1, 0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 97, 205, 448, 1004, 2339, 5661, 14291, 37507, 101962, 285386, 817772, 2386946, 7069893, 21195092, 64225525, 196636559, 608551084, 1905848637, 6049696252, 19501015441, 63960251538, 213822965681, 729536174204, 2541833303563
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OFFSET
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0,6
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LINKS
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FORMULA
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a(7) = 6: 12|3, 134|2, 13|24, 14|23, 1|234, 123456.
a(8) = 11: 124|3, 12|34, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1234567.
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MAPLE
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b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
`if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
end:
a:= n-> add(b(k, 1, 0, n-k), k=0..n):
seq(a(n), n=0..42);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A365903
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Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.
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+10
3
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1, 1, 1, 2, 4, 10, 29, 101, 367, 1562, 6891, 37871, 197930, 1121634, 6888085, 46190282, 323250987, 2349020516, 17897285514, 142512956148, 1178963284732, 10248806222398, 91421283039658, 847666112839362, 8100455404172267, 79925567946537362, 814508927747776069
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OFFSET
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0,4
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LINKS
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MAPLE
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b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
`if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
end:
a:= n-> max(seq(b(k, 1, 0, n), k=0..n*(n+1)/2)):
seq(a(n), n=0..26);
# second Maple program:
a:= proc(h) option remember; local b; b:=
proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^(h-n+1)*b(n-1, m+1)))
end: forget(b); max(coeffs(b(h, 0)))
end:
seq(a(n), n=0..26);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A368401
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Number T(n,k) of permutations of [n] whose sum of cycle maxima minus cycle minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows.
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+10
2
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1, 1, 1, 1, 1, 2, 3, 1, 3, 7, 11, 2, 1, 4, 12, 28, 53, 12, 10, 1, 5, 18, 52, 135, 289, 84, 72, 58, 6, 1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42, 1, 7, 33, 125, 429, 1407, 4545, 12983, 3520, 3976, 4292, 3950, 3422, 790, 486, 330, 24
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OFFSET
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0,6
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LINKS
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FORMULA
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EXAMPLE
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T(3,0) = 1: (1)(2)(3).
T(3,1) = 2: (12)(3), (1)(23).
T(3,2) = 3: (123), (132), (13)(2).
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 2, 3;
1, 3, 7, 11, 2;
1, 4, 12, 28, 53, 12, 10;
1, 5, 18, 52, 135, 289, 84, 72, 58, 6;
1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42;
...
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MAPLE
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b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
b(n-1, s)+add(b(n-1, subs(h=h+[0, 1], s)), h=s), 0)+
`if`(n>k+1, b(n-1, {s[], [n, 1]}), 0)+add(h[2]!*expand(
x^(h[1]-n)*b(n-1, s minus {h})), h=s))(nops(s)))
end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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