Displaying 1-10 of 11 results found.
Central coefficients of the polynomials defined in A278074.
+20
5
1, 1, 16510, 17651304000, 286988816206755000, 35284812773848049161035000, 21735699944364325706210750640600000, 51125456932397825107093888817556205542000000, 378603085421985456745667562645258531056443927230000000, 7641597761030055776217194099395682779700673105680593973250000000
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
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:
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))
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
COMMENTS
Also the central coefficients of the polynomials defined in A278073 for m = 2.
Central coefficients of the polynomials defined in A278073.
+10
5
1, 1, 1364, 42771456, 10298900437056, 11287986820196486400, 41397337338743872194508800, 414528538783792919989135797964800, 9808376038359632185170127842947907993600, 492228239722024416239987973400425228541016064000
Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).
+10
4
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
COMMENTS
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}
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.
REFERENCES
G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.
FORMULA
F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.
EXAMPLE
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]
MAPLE
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;
MATHEMATICA
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]];
PROG
(Sage)
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))
Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).
+10
3
1, 1, 0, 69, 1, 0, 33661, 988, 1, 0, 60376809, 2669683, 16507, 1, 0, 288294050521, 17033188586, 212734266, 261626, 1, 0, 3019098162602349, 223257353561605, 4297382231090, 17634518610, 4196345, 1, 0
FORMULA
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) for n>0 and F_{4; 0}(x) = 1.
EXAMPLE
Triangle starts:
[n\k][ 0 1 2 3 4 5]
--------------------------------------------------
[0] [ 1]
[1] [ 1, 0]
[2] [ 69, 1, 0]
[3] [ 33661, 988, 1, 0]
[4] [ 60376809, 2669683, 16507, 1, 0]
[5] [288294050521, 17033188586, 212734266, 261626, 1, 0]
MAPLE
Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292606_row := proc(n) if n = 0 then return [1] fi;
add( A278074(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end: for n from 0 to 6 do A292606_row(n) od;
PROG
if n == 0: return [1]
S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
return expand(S).list() + [0]
for n in (0..5): print( A292606_row(n))
Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 2 and 0 <= k <= n.
+10
3
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
FORMULA
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).
EXAMPLE
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]
MAPLE
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)]);
MATHEMATICA
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]&;
PROG
(Sage)
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))
Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 4 and 0 <= k <= n.
+10
3
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
FORMULA
T(n, k) = T_{4}(n, k) where T_{m}(n, k) is defined in A326477.
EXAMPLE
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]
Coefficients of polynomials related to ordered set partitions. Triangle read by rows, T_{m}(n, k) for m = 3 and 0 <= k <= n.
+10
3
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
FORMULA
T(n, k) = T_{3}(n, k) where T_{m}(n, k) is defined in A326477.
EXAMPLE
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]
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.
+10
3
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
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 = 4. For instance 4*P(4, .) = [[16], [12, 4], [8, 8], [8, 4, 4], [4, 4, 4, 4]].
EXAMPLE
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.
PROG
(Sage) # uses[GenOrdSetPart from A327022] def A327024row(n): return GenOrdSetPart(4, n)
for n in (0..6): print(A327024row(n))
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