A281480 -id:A281480 - cp's OEIS frontend

A278074 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 = 4.

1, 0, 1, 0, 1, 70, 0, 1, 990, 34650, 0, 1, 16510, 2702700, 63063000, 0, 1, 261630, 213519150, 17459442000, 305540235000, 0, 1, 4196350, 17651304000, 4350310965000, 231905038365000, 3246670537110000

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle starts:
[1]
[0, 1]
[0, 1,     70]
[0, 1,    990,     34650]
[0, 1,  16510,   2702700,    63063000]
[0, 1, 261630, 213519150, 17459442000, 305540235000]

Robert Israel, Table of n, a(n) for n = 0..8510

Cf. A014608 (diagonal), A243665 (row sums), A211212 (alternating row sums), A281480 (central coefficients).
Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278073 (m=3).
Cf. A327024 (refinement).

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(4,n), x) od;
# Alternatively:
A278074_row := proc(n) 1/(1-t*((cosh(x)+cos(x))/2-1)); expand(series(%,x,4*n+1));
(4*n)!*coeff(%,x,4*n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 5 do A278074_row(n) od;

With[{m = 4}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 24, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten

# uses [P from A278073]
def A278074_row(n): return list(P(4, n))
for n in (0..6): print(A278074_row(n)) # Peter Luschny, Mar 24 2020

E.g.f.: 1/(1-t*((cosh(x)+cos(x))/2-1)), 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);

A281479 Central coefficients of the polynomials defined in A278073.

Original entry on oeis.org

1, 1, 1364, 42771456, 10298900437056, 11287986820196486400, 41397337338743872194508800, 414528538783792919989135797964800, 9808376038359632185170127842947907993600, 492228239722024416239987973400425228541016064000

Offset: 0

Peter Luschny, Jan 22 2017

nonn

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(3,2*n),x,n): seq(A281479(n),n=0..9);

A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

Original entry on oeis.org

1, 1, 1, 70, 1, 990, 34650, 1, 3640, 12870, 2702700, 63063000, 1, 9690, 251940, 26453700, 187065450, 17459442000, 305540235000, 1, 21252, 1470942, 2704156, 154448910, 8031343320, 9465511770, 374796021600, 3975514943400, 231905038365000, 3246670537110000

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 = 4. For instance 4*P(4, .) = [[16], [12, 4], [8, 8], [8, 4, 4], [4, 4, 4, 4]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;    70]
[3] [1;   990;   34650]
[4] [1;  3640,   12870;  2702700;  63063000]
[5] [1;  9690,  251940; 26453700, 187065450; 17459442000; 305540235000]
[6] [1; 21252, 1470942,  2704156; 154448910,  8031343320,   9465511770;
     374796021600, 3975514943400; 231905038365000; 3246670537110000]
.
T(4, 1) = 3640 because [12, 4] is the integer partition 4*P(4, 1) in the canonical order and there are 1820 set partitions which have the shape [12, 4]. Finally, since the order of the sets is taken into account, one gets 2!*1820 = 3640.

Row sums: A243665, alternating row sums: A211212, main diagonal: A014608, central column: A281480, by length: A278074.
Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), A327023 (m=3), this sequence (m=4).
Cf. A080577, A000041, A072233.

# uses[GenOrdSetPart from A327022]
def A327024row(n): return GenOrdSetPart(4, n)
for n in (0..6): print(A327024row(n))

cp's OEIS Frontend

A278074 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 = 4.

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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

A281479 Central coefficients of the polynomials defined in A278073.

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Maple

A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, 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

A278074 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 = 4.

Views

Author

Keywords

Examples

Links

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

A281479 Central coefficients of the polynomials defined in A278073.

Views

Author

Keywords

Crossrefs

Programs

Maple

A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Sage

A278074 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 = 4.