A281478 -id:A281478 - cp's OEIS frontend

A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

1, 1, 6, 1, 30, 90, 1, 126, 1260, 2520, 1, 510, 13230, 75600, 113400, 1, 2046, 126720, 1580040, 6237000, 7484400, 1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400, 1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000, 1, 131070, 96461910, 8203431600, 196556560200, 1882311631200, 8266953895200, 16672848192000, 12504636144000

Offset: 1

N. J. A. Sloane, Apr 22 2014

T(n,k) gives the number of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. An example is given below. Cf. A019538 and A156289. - Peter Bala, Aug 20 2014

			Triangle begins:
1,
1, 6,
1, 30, 90,
1, 126, 1260, 2520,
1, 510, 13230, 75600, 113400,
1, 2046, 126720, 1580040, 6237000, 7484400,
1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400,
1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000,
...
From _Peter Bala_, Aug 20 2014: (Start)
Row 2: [1,6]
k  Ordered set partitions of {1,2,3,4} into k blocks    Number
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1   {1,2,3,4}                                             1
2   {1,2}{3,4}, {3,4}{1,2}, {1,3}{2,4}, {2,4}{1,3},       6
    {1,4}{2,3}, {2,3}{1,4}
(End)

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
S. A. Joffe, Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 48 (1917-1920), 193-271.

Muniru A Asiru, Rows n = 1..50 of triangle, flattened

Case m=2 of the polynomials defined in A278073.
Cf. A000680 (diagonal), A094088 (row sums), A000364 (alternating row sums), A281478 (central terms), A327022 (refinement).
Diagonals give A002446, A213455, A241172, A002456.
Cf. A019538, A036969, A156289.

Flat(List([1..10],n->List([1..n],k->1/(2^(k-1))*Sum([1..k],j->(-1)^(k-j)*Binomial(2*k,k-j)*j^(2*n))))); # Muniru A Asiru, Feb 27 2019

T := proc(n,k) option remember;
if k > n then 0
elif k=0 then k^n
elif k=1 then 1
else k*(2*k-1)*T(n-1,k-1)+k^2*T(n-1,k); fi;
end: # Minor edit to make it also work in the (0,0)-offset case. Peter Luschny, Sep 03 2022
for n from 1 to 12 do lprint([seq(T(n,k), k=1..n)]); od:

T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[, 1] = 1; T[, ] = 0; Table[T[n, k], {n, 1, 9}, {k, 1, n}] (* _Jean-François Alcover, Jul 03 2019 *)

@cached_function
def A241171(n, k):
    if n == 0 and k == 0: return 1
    if k < 0 or k > n: return 0
    return (2*k^2 - k)*A241171(n - 1, k - 1) + k^2*A241171(n - 1, k)
for n in (1..6): print([A241171(n, k) for k in (1..n)]) # Peter Luschny, Sep 06 2017

T(n,k) = 0 if k <= 0 or k > n, = 1 if k=1, otherwise T(n,k) = k*(2*k-1)*T(n-1,k-1) + k^2*T(n-1,k).
Related to Euler numbers A000364 by A000364(n) = (-1)^n*Sum_{k=1..n} (-1)^k*T(n,k). For example, A000364(3) = 61 = 90 - 30 + 1.
From Peter Bala, Aug 20 2014: (Start)
T(n,k) = 1/(2^(k-1))*Sum_{j = 1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).
T(n,k) = k!*A156289(n,k) = k!*(2*k-1)!!*A036969.
E.g.f.: A(t,z) := 1/( 1 - t*(cosh(z) - 1) ) = 1 + t*z^2/2! + (t + 6*t^2)*z^4/4! + (t + 30*t^2 + 90*t^3)*z^6/6! + ... satisfies the partial differential equation d^2/dz^2(A) = D(A), where D = t^2*(2*t + 1)*d^2/dt^2 + t*(5*t + 1)*d/dt + t.
Hence the row polynomials R(n,t) satisfy the differential equation R(n+1,t) = t^2*(2*t + 1)*R''(n,t) + t*(5*t + 1)*R'(n,t) + t*R(n,t) with R(0,t) = 1, where ' indicates differentiation w.r.t. t. This is equivalent to the above recurrence equation.
Recurrence for row polynomials: R(n,t) = t*( Sum_{k = 1..n} binomial(2*n,2*k)*R(n-k,t) ) with R(0,t) := 1.
Row sums equal A094088(n) for n >= 1.
A100872(n) = (1/2)*R(n,2). (End)

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);

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);

A327022 Partition triangle read by rows. Number of ordered set partitions of the set {1, 2, ..., 2*n} with all block sizes divisible by 2.

Original entry on oeis.org

1, 1, 1, 6, 1, 30, 90, 1, 56, 70, 1260, 2520, 1, 90, 420, 3780, 9450, 75600, 113400, 1, 132, 990, 924, 8910, 83160, 34650, 332640, 1247400, 6237000, 7484400, 1, 182, 2002, 6006, 18018, 270270, 252252, 630630, 1081080, 15135120, 12612600, 37837800, 189189000, 681080400, 681080400

Offset: 0

Peter Luschny, Aug 27 2019

We call an irregular triangle T a partition triangle if T(n, k) is defined for n >= 0 and 0 <= k < A000041(n).

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 = 2. For instance 2*P(4, .) = [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;   6]
[3] [1;  30;  90]
[4] [1;  56,  70; 1260; 2520]
[5] [1;  90, 420; 3780, 9450; 75600; 113400]
[6] [1; 132, 990,  924; 8910, 83160,  34650; 332640, 1247400; 6237000; 7484400]
.
T(4, 1) = 56 because [6, 2] is the integer partition 2*P(4, 1) in the canonical order and there are 28 set partitions which have the shape [6, 2] (an example is {{1, 3, 4, 5, 6, 8}, {2, 7}}). Finally, since the order of the sets is taken into account, one gets 2!*28 = 56.

Row sums: A094088, alternating row sums: A028296, main diagonal: A000680, central column A281478, by length: A241171.
Cf. A178803 (m=0), A133314 (m=1), this sequence (m=2), A327023 (m=3), A327024 (m=4).
Cf. A080577, A000041, A072233.

def GenOrdSetPart(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return [factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes]
def A327022row(n): return GenOrdSetPart(2, n)
for n in (0..6): print(A327022row(n))

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A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

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A281479 Central coefficients of the polynomials defined in A278073.

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A281480 Central coefficients of the polynomials defined in A278074.

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Maple

A327022 Partition triangle read by rows. Number of ordered set partitions of the set {1, 2, ..., 2*n} with all block sizes divisible by 2.

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