A366942
Expansion of e.g.f. 1/(1-x-2*x^2-3*x^3).
Original entry on oeis.org
1, 1, 6, 48, 408, 5040, 72000, 1184400, 22619520, 482993280, 11459750400, 299495750400, 8531976499200, 263353163673600, 8754879893760000, 311808414677760000, 11845876873678848000, 478163414336864256000, 20436460099541950464000, 921972301728418676736000
Offset: 0
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a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-j)*binomial(n, j)*j!*j, j=1..min(3, n)))
end:
seq(a(n), n=0..19); # Alois P. Heinz, Dec 14 2023
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With[{m = 20}, Range[0, m]! * CoefficientList[Series[1/(1 - x - 2*x^2 - 3*x^3), {x, 0, m}], x]] (* Amiram Eldar, Oct 30 2023 *)
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my(x='x+O('x^25)); Vec(serlaplace(1/(1-x-2*x^2-3*x^3))) \\ Michel Marcus, Oct 30 2023
A366950
Expansion of e.g.f. exp(x^2+3*x^3).
Original entry on oeis.org
1, 0, 2, 18, 12, 360, 3360, 7560, 183120, 1814400, 8195040, 184615200, 1976546880, 14166472320, 310589959680, 3634245014400, 36092331475200, 787170153369600, 10123771065408000, 127736406404006400, 2807613032557132800, 39732753299855616000
Offset: 0
a(6)=3360 since the number of ways to partition [6] into unordered blocks of size at most 3, order the elements within each block, and select 2 elements from each block are the following:
1) 12,34,56: 120 ways to order elements in unordered blocks, 1 way to choose 2 elements from each block, hence 120 ways;
2) 123,456: 360 ways to order elements in unordered blocks, 3*3 ways to choose 2 elements from each block, hence 3240 ways.
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With[{m = 21}, Range[0, m]! * CoefficientList[Series[Exp[x^2 + 3*x^3], {x, 0, m}], x]] (* Amiram Eldar, Oct 30 2023 *)
A365293
a(n) = n!*tetranacci(n+3).
Original entry on oeis.org
1, 1, 4, 24, 192, 1800, 20880, 282240, 4354560, 75479040, 1455148800, 30855686400, 713712384000, 17884003737600, 482619020083200, 13954193180928000, 430360865206272000, 14102295149150208000, 489295008086556672000, 17919783031425859584000
Offset: 0
a(5) = 1800 since the number of ways to partition [5] into blocks of size at most 4, order the blocks, and order the elements within each block are the following:
1) 1234,5: 10 such ordered blocks; 240 ways;
2) 123,4,5: 60 such ordered blocks; 360 ways;
3) 123,45: 20 such ordered blocks; 240 ways;
4) 12,34,5: 90 such ordered blocks; 360 ways;
5) 12,3,4,5: 240 such ordered blocks; 480 ways;
6) 1,2,3,4,5: 120 such ordered blocks; 120 ways.
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Table[n! SeriesCoefficient[1/(1-x-x^2-x^3-x^4),{x,0,n}],{n,0,19}] (* Stefano Spezia, Aug 31 2023 *)
A365330
Expansion of e.g.f. x^3/(1-x-x^2-x^3)^2.
Original entry on oeis.org
0, 0, 0, 6, 48, 600, 8640, 131040, 2257920, 42819840, 885427200, 19918483200, 483791616000, 12622171161600, 352200296448000, 10466625641472000, 330077933273088000, 11010660024139776000, 387369218691366912000, 14335266857678807040000, 556691771706962411520000
Offset: 0
a(6)=8640 since the ways to partition [6] into blocks of size at most 3, order the blocks, order the elements within each block, and select 3 elements from a block are the following:
(i) 123,4,5,6: 2880 such orderings, 1 way to choose three elements (from the block with 3 elements), hence 2880 ways;
(ii) 123,45,6: 4320 such orderings, 1 way to choose three elements (from the block with 3 elements), hence 4320 ways;
(iii) 123,456: 720 such orderings, 2 ways to choose three elements (from one of the two blocks with 3 elements), hence 1440 ways.
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With[{m = 20}, Range[0, m]! * CoefficientList[Series[x^3/(1 - x - x^2 - x^3)^2, {x, 0, m}], x]] (* Amiram Eldar, Sep 02 2023 *)
A366957
Expansion of e.g.f. 1/(1-x^2-3*x^3).
Original entry on oeis.org
1, 0, 2, 18, 24, 720, 7200, 45360, 1128960, 14152320, 199584000, 4909766400, 82388275200, 1793381990400, 47163455539200, 1051370191872000, 29396519792640000, 863253387988992000, 24437860434763776000, 807966756915462144000, 27000346486744350720000
Offset: 0
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With[{m = 20}, Range[0, m]! * CoefficientList[Series[1/(1 - x^2 - 3*x^3), {x, 0, m}], x]] (* Amiram Eldar, Oct 30 2023 *)
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