A281479 -id:A281479

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A278073

Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

+10
18

1, 0, 1, 0, 1, 20, 0, 1, 168, 1680, 0, 1, 1364, 55440, 369600, 0, 1, 10920, 1561560, 33633600, 168168000, 0, 1, 87380, 42771456, 2385102720, 34306272000, 137225088000, 0, 1, 699048, 1160164320, 158411809920, 5105916816000, 54752810112000, 182509367040000

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..35.

FORMULA

E.g.f.: 1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1)), nonzero terms.

EXAMPLE

Triangle begins:

[1]

[0, 1]

[0, 1, 20]

[0, 1, 168, 1680]

[0, 1, 1364, 55440, 369600]

[0, 1, 10920, 1561560, 33633600, 168168000]

MAPLE

P := proc(m, n) option remember; if n = 0 then 1 else

add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:

for n from 0 to 6 do PolynomialTools:-CoefficientList(P(3, n), x) od;

# Alternatively:

A278073_row := proc(n)

1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1));

expand(series(%, x, 3*n+1)); (3*n)!*coeff(%, x, 3*n);

PolynomialTools:-CoefficientList(%, t) end:

for n from 0 to 6 do A278073_row(n) od;

MATHEMATICA

With[{m = 3}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 21, m}]];

Function[arg, CoefficientList[arg, t]] /@ % // Flatten

PROG

(Sage)

R = PowerSeriesRing(ZZ, 'x')

x = R.gen().O(30)

@cached_function

def P(m, n):

if n == 0: return R(1)

return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))

def A278073_row(n): return list(P(3, n))

for n in (0..6): print(A278073_row(n)) # Peter Luschny, Mar 24 2020

CROSSREFS

Cf. A014606 (diagonal), A243664 (row sums), A002115 (alternating row sums), A281479 (central coefficients), A327023 (refinement).

Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278074 (m=4).

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Jan 22 2017

STATUS

approved

A281478

Central coefficients of Joffe's central differences of zero (assuming offset 0 and T(n,k) extended to 0 <= k <= n in A241171).

+10
5

1, 1, 126, 126720, 494053560, 5283068427000, 126301275727704000, 5896518025761483120000, 488276203972584492344880000, 66735969985432035804226510800000, 14236685931434801591697761172512160000, 4533351707244550464920840944132383960960000, 2077486542875366717627638783543223150778585600000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Also the central coefficients of the polynomials defined in A278073 for m = 2.

LINKS

Table of n, a(n) for n=0..12.

MAPLE

# Function P defined in A278073.

A281479 := n -> coeff(P(2, 2*n), x, n): seq(A281479(n), n=0..9);

CROSSREFS

Cf. Central coefficients: A088218 (m=0), A210029 (m=1), A281478 (m=2), A281479 (m=3), A281480 (m=4). Related triangles: A097805 (m=0), A131689 (m=1), A241171 (m=2), A278073 (m=3), A278074 (m=4).

KEYWORD

nonn

AUTHOR

Peter Luschny, Jan 22 2017

STATUS

approved

A281480

Central coefficients of the polynomials defined in A278074.

+10
5

1, 1, 16510, 17651304000, 286988816206755000, 35284812773848049161035000, 21735699944364325706210750640600000, 51125456932397825107093888817556205542000000, 378603085421985456745667562645258531056443927230000000, 7641597761030055776217194099395682779700673105680593973250000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Robert Israel, Table of n, a(n) for n = 0..70

MAPLE

# Function P defined in A278074.

A281480 := n -> coeff(P(4, 2*n), x, n): seq(A281480(n), n=0..9);

CROSSREFS

Central coefficients: A088218 (m=0), A210029 (m=1), A281478 (m=2), A281479 (m=3), A281480 (m=4). Related triangles: A097805 (m=0), A131689 (m=1), A241171 (m=2), A278073 (m=3), A278074 (m=4).

KEYWORD

nonn

AUTHOR

Peter Luschny, Jan 22 2017

STATUS

approved

A327023

Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

+10
3

1, 1, 1, 20, 1, 168, 1680, 1, 440, 924, 55440, 369600, 1, 910, 10010, 300300, 1261260, 33633600, 168168000, 1, 1632, 37128, 48620, 1113840, 24504480, 17153136, 326726400, 2058376320, 34306272000, 137225088000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 3. For instance 3*P(4, .) = [[12], [9, 3], [6, 6], [6, 3, 3], [3, 3, 3, 3]].

LINKS

Table of n, a(n) for n=0..29.

EXAMPLE

Triangle starts (note the subdivisions by ';' (A072233)):

[0] [1]

[1] [1]

[2] [1; 20]

[3] [1; 168; 1680]

[4] [1; 440, 924; 55440; 369600]

[5] [1; 910, 10010; 300300, 1261260; 33633600; 168168000]

[6] [1; 1632, 37128, 48620; 1113840, 24504480, 17153136; 326726400, 2058376320;

34306272000; 137225088000]

T(4, 1) = 440 because [9, 3] is the integer partition 3*P(4, 1) in the canonical order and there are 220 set partitions which have the shape [9, 3]. Finally, since the order of the sets is taken into account, one gets 2!*220 = 440.

PROG

(Sage) # uses[GenOrdSetPart from A327022]

def A327023row(n): return GenOrdSetPart(3, n)

for n in (0..6): print(A327023row(n))

CROSSREFS

Row sums: A243664, alternating row sums: A002115, main diagonal: A014606, central column A281479, by length: A278073.

Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), this sequence (m=3), A327024 (m=4).

Cf. A080577, A000041, A072233.

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Aug 27 2019

STATUS

approved

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