A241171 -id:A241171 - cp's OEIS frontend

A281479 Central coefficients of the polynomials defined in A278073.

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);

A088874 T(n, k) = [x^k] (2n)! [z^(2n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 16, 30, 15, 0, 272, 588, 420, 105, 0, 7936, 18960, 16380, 6300, 945, 0, 353792, 911328, 893640, 429660, 103950, 10395, 0, 22368256, 61152000, 65825760, 36636600, 11351340, 1891890, 135135, 0, 1903757312

Offset: 0

Philippe Deléham, Nov 26 2003

Previous name was: Triangle read by rows, given by [0, 2, 6, 12, 20, 30, 42, 56, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, ...] where Delta is the operator defined in A084938.

			Triangle starts:
[0] 1
[1] 0, 1
[2] 0, 2,        3
[3] 0, 16,       30,       15
[4] 0, 272,      588,      420,      105
[5] 0, 7936,     18960,    16380,    6300,     945
[6] 0, 353792,   911328,   893640,   429660,   103950,   10395
[7] 0, 22368256, 61152000, 65825760, 36636600, 11351340, 1891890, 135135

Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) #09.7.6.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles , JIS 9 (2006) #06.4.1.
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.

Another version of the triangle A085734. A signed version is A318146.

Diagonals give: A000007 A000182 A001147, row sums A000364.

ser := series(sec(z)^x, z, 24): row := n -> n!*coeff(ser, z, n):
seq(seq(coeff(row(2*n), x, k), k=0..n), n=0..8); # Peter Luschny, Jul 01 2019

T[1, 1] = 1; T[n_, k_] := Sum[(1/2^(j-1))*StirlingS1[j, k-1]*Sum[(-1)^(i + k + n)*(i-j)^(2(n-1)) Binomial[2j, i], {i, 0, j-1}]/j!, {j, 1, n-1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018, after Vladimir Kruchinin *)
a[n_] := (2n)! SeriesCoefficient[Sec[z]^x, {z, 0, 2n}] // CoefficientList[#, x] &;
Table[a[n], {n, 0, 8}] // Flatten (* Peter Luschny, Jul 01 2019 *)

# uses [A241171]
def fr2_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    L = expand(S).list()
    return sum(L[k]*binomial(x+k, n) for k in (0..n-1)).list()
A088874_row = lambda n: [(-1)^(n-k)*m for k,m in enumerate(fr2_row(n))]
for n in (0..7): print(A088874_row(n)) # Peter Luschny, Sep 19 2017

T(n, k) = A085734(n-1, k-1) for n>0 and k>0.

T(n, k) = [x^k] (2*n)! [z^(2*n)] sec(z)^x. - Peter Luschny, Jul 01 2019

New name by Peter Luschny, Jul 01 2019

A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

Original entry on oeis.org

1, 1, 0, 5, 1, 0, 61, 28, 1, 0, 1385, 1011, 123, 1, 0, 50521, 50666, 11706, 506, 1, 0, 2702765, 3448901, 1212146, 118546, 2041, 1, 0, 199360981, 308869464, 147485535, 24226000, 1130235, 8184, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:
F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.
F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.
F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.
F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.
The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.
Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:
F_{m} :   F_{0}    F_{1}    F_{2}    F_{3}    F_{4}
x = -1:  A165326  A155585  A002105  A292609  A292607
x =  1:  A000012  A000142  A000680  A014606  A014608  ... (m*n)!/m!^n
x =  0:    --     A000012  A000364  A002115  A211212  ... m-alternating permutations of length m*n.
Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.

			Triangle starts:
[n\k][    0        1        2       3     4  5  6]
--------------------------------------------------
[0][      1]
[1][      1,       0]
[2][      5,       1,       0]
[3][     61,      28,       1,      0]
[4][   1385,    1011,     123,      1,    0]
[5][  50521,   50666,   11706,    506,    1, 0]
[6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]

G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.

F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.
First column: A000364. Row sums: A000680. Alternating row sums: A002105.
Cf. A181985, A241171.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292604_row := proc(n) if n = 0 then return [1] fi;
add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292604_row(n) od;

T[n_, k_] /; 1 <= k <= n := T[n, k] = k (2 k - 1) T[n - 1, k - 1] + k^2 T[n - 1, k]; T[, 1] = 1; T[, _] = 0;
F[2, 0][] = 1; F[2, n][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
Table[row[n], {n, 0, 7}] (* Jean-François Alcover, Jul 06 2019 *)

def A292604_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..6): print(A292604_row(n))

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.

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))

A326477 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 4, 3, 0, 46, 60, 15, 0, 1114, 1848, 840, 105, 0, 46246, 88770, 54180, 12600, 945, 0, 2933074, 6235548, 4574130, 1469160, 207900, 10395, 0, 263817646, 605964450, 505915410, 199849650, 39729690, 3783780, 135135

Offset: 0

Peter Luschny, Jul 08 2019

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 4, 3]
[3] [0, 46, 60, 15]
[4] [0, 1114, 1848, 840, 105]
[5] [0, 46246, 88770, 54180, 12600, 945]
[6] [0, 2933074, 6235548, 4574130, 1469160, 207900, 10395]

Row sums A094088. Alternating row sums A153881 starting at 0.
Main diagonal A001147. Associated set partitions A241171.
A129062 (m=1, associated with A131689), this sequence (m=2), A326587 (m=3, associated with A278073), A326585 (m=4, associated with A278074).

CL := f -> PolynomialTools:-CoefficientList(f, x):
FL := s -> ListTools:-Flatten(s, 1):
StirPochConv := proc(m, n) local P, L; P := proc(m, n) option remember;
`if`(n = 0, 1, add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n)) end:
L := CL(P(m, n)); CL(expand(add(L[k+1]*pochhammer(x,k)/k!, k=0..n))) end:
FL([seq(StirPochConv(2,n), n = 0..7)]);

P[, 0] = 1; P[m, n_] := P[m, n] = Sum[Binomial[m*n, m*k]*P[m, n-k]*x, {k, 1, n}] // Expand;
T[m_][n_] := CoefficientList[P[m, n], x].Table[Pochhammer[x, k]/k!, {k, 0, n}] // CoefficientList[#, x]&;
Table[T[2][n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jul 21 2019 *)

def StirPochConv(m, n):
    z = var('z'); R = ZZ[x]
    F = [i/m for i in (1..m-1)]
    H = hypergeometric([], F, (z/m)^m)
    P = R(factorial(m*n)*taylor(exp(x*(H-1)), z, 0, m*n + 1).coefficient(z, m*n))
    L = P.list()
    S = sum(L[k]*rising_factorial(x,k) for k in (0..n))
    return expand(S).list()
for n in (0..6): print(StirPochConv(2, n))

For m >= 1 let P(m,0) = 1 and P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)*P(m, n-k)*x for n > 0. Then T_{m}(n, k) = Sum_{k=0..n} ([x^k]P(m, n))*rf(x,k)/k! where rf(x,k) are the rising factorial powers. T(n, k) = T_{2}(n, k).

A326585 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 4 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 36, 35, 0, 12046, 17820, 5775, 0, 16674906, 30263480, 16216200, 2627625, 0, 65544211366, 135417565890, 93516348900, 26189163000, 2546168625, 0, 588586227465426, 1334168329550300, 1083314031995250, 402794176785000, 69571511509500, 4509264634875

Offset: 0

Peter Luschny, Jul 21 2019

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 36, 35]
[3] [0, 12046, 17820, 5775]
[4] [0, 16674906, 30263480, 16216200, 2627625]
[5] [0, 65544211366, 135417565890, 93516348900, 26189163000, 2546168625]
[6] [0, 588586227465426, 1334168329550300, 1083314031995250, 402794176785000, 69571511509500, 4509264634875]

Row sums A243665. Main diagonal A025036.

A129062 (m=1, associated with A131689), A326477 (m=2, associated with A241171), A326587 (m=3, associated with A278073), this sequence (m=4, associated with A278074).

Maple
```
# See A326477.
```

T(n, k) = T_{4}(n, k) where T_{m}(n, k) is defined in A326477.

A326587 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 3 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 11, 10, 0, 645, 924, 280, 0, 111563, 197802, 101640, 15400, 0, 42567981, 86271640, 57717660, 15415400, 1401400, 0, 30342678923, 67630651098, 53492240256, 19158419280, 3144741600, 190590400

Offset: 0

Peter Luschny, Jul 20 2019

			Triangle starts:
0 [1]
1 [0, 1]
2 [0, 11, 10]
3 [0, 645, 924, 280]
4 [0, 111563, 197802, 101640, 15400]
5 [0, 42567981, 86271640, 57717660, 15415400, 1401400]
6 [0, 30342678923, 67630651098, 53492240256, 19158419280, 3144741600, 190590400]

Row sums A243664. Main diagonal A025035.

A129062 (m=1, associated with A131689), A326477 (m=2, associated with A241171), this sequence (m=3, associated with A278073), A326585 (m=4, associated with A278074).

Maple
```
# See A326477.
```

T(n, k) = T_{3}(n, k) where T_{m}(n, k) is defined in A326477.

A213455 90*A002451(n).

Original entry on oeis.org

90, 1260, 13230, 126720, 1171170, 10663380, 96461910, 870123240, 7838973450, 70582218300, 635365793790, 5718795460560, 51471172410930, 463248604762020, 4169269655112870, 37523555745034680, 337712517101387610, 3039414715496790540, 27354740685808323150, 246192699157623741600, 2215734424360009007490

Offset: 0

N. J. A. Sloane, Jun 12 2012

nonn

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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Index entries for linear recurrences with constant coefficients, signature (14, -49, 36).

Cf. A002451. A diagonal of the triangle in A241171.

From Chai Wah Wu, Jan 30 2018: (Start)
a(n) = 14*a(n-1) - 49*a(n-2) + 36*a(n-3) for n > 2.
G.f.: 90/((1 - x)*(1 - 4*x)*(1 - 9*x)). (End)

A318259 Generalized Worpitzky numbers W_{m}(n,k) for m = 2, n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, -1, 1, 5, -11, 6, -61, 211, -240, 90, 1385, -6551, 11466, -8820, 2520, -50521, 303271, -719580, 844830, -491400, 113400, 2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400, -199360981, 1704396331, -6187282920, 12372329970, -14727913200, 10443232800, -4086482400, 681080400

Offset: 0

Peter Luschny, Sep 06 2018

The triangle can be seen as a member of a family of generalized Worpitzky numbers A028246. See the cross-references for some other members.

The unsigned numbers have row sums A210657 which points to an interpretation of the unsigned numbers as a refinement of marked Schröder paths (see Josuat-Vergès and Kim).

			[0] [      1]
[1] [     -1,         1]
[2] [      5,       -11,        6]
[3] [    -61,       211,     -240,        90]
[4] [   1385,     -6551,    11466,     -8820,     2520]
[5] [ -50521,    303271,  -719580,    844830,  -491400,    113400]
[6] [2702765, -19665491, 58998126, -93511440, 82661040, -38669400, 7484400]

Matthieu Josuat-Vergès and Jang Soo Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, arXiv:1101.5608 [math.CO], 2011.

Row sums are A000007, alternating row sums are A210657.
Cf. T(n,n) = A000680, T(n, 0) = A028296(n) (Gudermannian), A000364 (Euler secant), A241171 (Joffe's differences), A028246 (Worpitzky).
Cf. A167374 (m=0), A028246 & A163626 (m=1), this seq (m=2), A318260 (m=3).

Joffe := proc(n, k) option remember; if k > n then 0 elif k = 0 then k^n else
k*(2*k-1)*Joffe(n-1, k-1)+k^2*Joffe(n-1, k) fi end:
T := (n, k) -> add((-1)^(k-j)*binomial(n-j, n-k)*add((-1)^i*Joffe(n,i)*
binomial(n-i, j), i=0..n), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..6);

Joffe[0, 0] = 1; Joffe[n_, k_] := Joffe[n, k] = If[k>n, 0, If[k == 0,k^n, k*(2*k-1)*Joffe[n-1, k-1] + k^2*Joffe[n-1, k]]];
T[n_, k_] := Sum[(-1)^(k-j)*Binomial[n-j, n-k]*Sum[(-1)^i*Joffe[n, i]* Binomial[n-i, j], {i, 0, n}], {j, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2019, from Maple *)

def EW(m, n):
    @cached_function
    def S(m, n):
        R. = ZZ[]
        if n == 0: return R(1)
        return R(sum(binomial(m*n, m*k)*S(m, n-k)*x for k in (1..n)))
    s = S(m, n).list()
    c = lambda k: sum((-1)^(k-j)*binomial(n-j,n-k)*
        sum((-1)^i*s[i]*binomial(n-i,j) for i in (0..n)) for j in (0..k))
    return [c(k) for k in (0..n)]
def A318259row(n): return EW(2, n)
flatten([A318259row(n) for n in (0..6)])

Let S(n, k) denote Joffe's central differences of zero (A241171) extended to the case n = 0 and k = 0 by prepending a column 1, 0, 0, 0,... to the triangle, then:

T(n,k) = Sum_{j=0..k}((-1)^(k-j)*C(n-j,n-k)*Sum_{i=0..n}((-1)^i*S(n,i)*C(n-i,j))).

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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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A088874 T(n, k) = [x^k] (2*n)! [z^(2*n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.

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A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

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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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A326477 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.

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A326585 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 4 and 0 <= k <= n.

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A326587 Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 3 and 0 <= k <= n.

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A088874 T(n, k) = [x^k] (2n)! [z^(2n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.