A281479 -id:A281479 - cp's OEIS frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

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

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle begins:
[1]
[0, 1]
[0, 1,    20]
[0, 1,   168,    1680]
[0, 1,  1364,   55440,   369600]
[0, 1, 10920, 1561560, 33633600, 168168000]

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).

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;

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

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

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.

A281478 Central coefficients of Joffe's central differences of zero (assuming offset 0 and T(n,k) extended to 0 <= k <= n in A241171).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jan 22 2017

nonn

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

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).

# Function P defined in A278073.
A281479 := n -> coeff(P(2, 2*n), x, n): seq(A281479(n), n=0..9);

A281480 Central coefficients of the polynomials defined in A278074.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Jan 22 2017

nonn

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

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).

# Function P defined in A278074.
A281480 := n -> coeff(P(4,2*n),x,n): seq(A281480(n),n=0..9);

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.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Aug 27 2019

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]].

			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.

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.

# uses[GenOrdSetPart from A327022]
def A327023row(n): return GenOrdSetPart(3, n)
for n in (0..6): print(A327023row(n))

cp's OEIS Frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

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A281478 Central coefficients of Joffe's central differences of zero (assuming offset 0 and T(n,k) extended to 0 <= k <= n in A241171).

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Maple

A281480 Central coefficients of the polynomials defined in A278074.

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Maple

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.

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Author

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Sage

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A281478 Central coefficients of Joffe's central differences of zero (assuming offset 0 and T(n,k) extended to 0 <= k <= n in A241171).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

A281480 Central coefficients of the polynomials defined in A278074.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.