A326659
T(n,k) = [0=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.
Original entry on oeis.org
1, 1, 1, 1, 4, 2, 1, 15, 18, 6, 1, 64, 132, 96, 24, 1, 325, 980, 1140, 600, 120, 1, 1956, 7830, 12720, 10440, 4320, 720, 1, 13699, 68502, 143850, 162120, 103320, 35280, 5040, 1, 109600, 657608, 1698816, 2447760, 2123520, 1108800, 322560, 40320
Offset: 0
Triangle T(n,k) begins:
1;
1, 1;
1, 4, 2;
1, 15, 18, 6;
1, 64, 132, 96, 24;
1, 325, 980, 1140, 600, 120;
1, 1956, 7830, 12720, 10440, 4320, 720;
1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
...
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T:= proc(n, k) option remember;
`if`(0=0, 1, 0)
end:
seq(seq(T(n, k), k=0..n), n=0..10);
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T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)
A327606
Expansion of e.g.f. exp(x)*(1-x)*x/(1-2*x)^2.
Original entry on oeis.org
0, 1, 8, 69, 712, 8705, 123456, 1994293, 36163184, 727518177, 16081980760, 387499155461, 10108673620728, 283851555270049, 8536572699232592, 273759055527114165, 9325469762472018016, 336282091434597013313, 12797935594025234906664, 512609204063389138693957
Offset: 0
-
a:= n-> n!*coeff(series(exp(x)*(1-x)*x/(1-2*x)^2, x, n+1), x, n):
seq(a(n), n=0..23);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, n^3,
2*(n+2)*a(n-1)-(4*n-1)*a(n-2)+2*(n-2)*a(n-3))
end:
seq(a(n), n=0..23);
-
With[{nn=20},CoefficientList[Series[Exp[x](1-x)(x/(1-2x)^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 15 2020 *)
A386374
Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs at least as many times as any other letter.
Original entry on oeis.org
1, 1, 3, 10, 47, 276, 2022, 17606, 179391, 2093860, 27581888, 404680398, 6541528886, 115437202986, 2206844818622, 45408726154590, 1000134868827263, 23468606700087972, 584340284516996400, 15383829737201853518, 426915367401366308112, 12454073547413511363878
Offset: 0
a(3) = 10 counts: (1,1,1), (1,1,2), (1,2,1), (1,2,3), (1,3,2), (2,1,1), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
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b:= proc(n, t) option remember; `if`(n=0, 1,
add(b(n-j, t)/j!, j=1..min(n, t)))
end:
a:= n-> n!*add(b(n-j, j)/j!, j=0..n):
seq(a(n), n=0..21); # Alois P. Heinz, Jul 19 2025
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A_x(N) = {my(x='x+O('x^N)); Vec(serlaplace(sum(i=0,N, x^i/(i! *(1-sum(j=1,i, x^j/j!))))))}
A386375
Number of words of length n over an infinite alphabet such that the letters cover an initial interval and the letter 1 occurs more frequently than any other letter.
Original entry on oeis.org
1, 1, 1, 4, 17, 96, 652, 5356, 51361, 568840, 7157036, 101048454, 1582644956, 27224336244, 509883010652, 10319902635984, 224283040843745, 5205554049801528, 128430045368430484, 3354764715348964222, 92460461868234201532, 2680680433302859375630, 81542551486359310209666
Offset: 0
a(5) = 96 counts the following words (number of permutations shown in brackets): (1,1,1,1,1) [1], (1,1,1,1,2) [5], (1,1,1,2,2) [10], (1,1,1,2,3) [20], (1,1,2,3,4) [60].
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b:= proc(n, t) option remember; `if`(n=0, 1,
add(b(n-j, t)/j!, j=1..min(n, t)))
end:
a:= n-> n!*add(b(n-j, j-1)/j!, j=0..n):
seq(a(n), n=0..22); # Alois P. Heinz, Jul 19 2025
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B_x(N) = {my(x='x+O('x^N)); Vec(serlaplace( sum(i=0,N, x^i/(i!*(1-sum(j=1,i-1, x^j/j!))))))}
Showing 1-4 of 4 results.
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