A291449 -id:A291449 - cp's OEIS frontend

A291447 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = BernoulliMedian(n).

0, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 4, 0, 0, 0, 1, -3, 48, -12, 36, 0, 0, 0, 1, -7, 268, -176, 1968, -216, 64, 0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400, 0, 0, 0, 1, -31, 4924, -11680, 488640, -238680, 496320, -639360, 5486400, -216000, 518400

Offset: 0

Peter Luschny, Aug 24 2017

The Bernoulli median numbers are A212196/A181131. See A290694 for further comments.

			Triangle starts:
[0, 1]
[0, 0, 0, 1]
[0, 0, 0, 1, -1, 4]
[0, 0, 0, 1, -3, 48, -12, 36]
[0, 0, 0, 1, -7, 268, -176, 1968, -216, 64]
[0, 0, 0, 1, -15, 240, -1580, 37140, -9900, 10400, -5760, 14400]
The first few polynomials are:
P_0(x) = x.
P_1(x) = (1/3)*x^3.
P_2(x) = (4/5)*x^5 - x^4 + (1/3)*x^3.
P_3(x) = (36/7)*x^7 - 12*x^6 + (48/5)*x^5 - 3*x^4 + (1/3)*x^3.
P_4(x) = 64*x^9 - 216*x^8 + (1968/7)*x^7 - 176*x^6 + (268/5)*x^5 - 7*x^4 +(1/3)*x^3.
Evaluated at x = 1 this gives a decomposition of the Bernoulli median numbers:
BM(0) = 1     =    1.
BM(1) = 1/3   =  1/3.
BM(2) = 2/15  =  4/5 -   1 +    1/3.
BM(3) = 8/105 = 36/7 -  12 +   48/5 -   3 +   1/3.
BM(4) = 8/105 =   64 - 216 + 1968/7 - 176 + 268/5 - 7 + 1/3.

Peter Luschny, Illustrating A291447

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

# The function BG_row is defined in A290694.
seq(BG_row(2, n, "num", "val"), n=0..12);        # A212196
seq(BG_row(2, n, "den", "val"), n=0..12);        # A181131
seq(print(BG_row(2, n, "num", "poly")), n=0..7); # A291447 (this seq.)
seq(print(BG_row(2, n, "den", "poly")), n=0..9); # A291448

T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j, 0, n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Numerator;
Table[Trow[r], {r, 0, 6}] // Flatten

T(n,k) = Numerator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

A291448 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n,x) such that Integral_{x=0..1} P'(n,x) = BernoulliMedian(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1

Offset: 0

Peter Luschny, Aug 24 2017

See A291447 and A290694 for comments.

			Triangle starts:
[1, 1]
[1, 1, 1, 3]
[1, 1, 1, 3, 1, 5]
[1, 1, 1, 3, 1, 5, 1, 7]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1]
[1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 11]
[1, 1, 1, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13]

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Maple
```
# See A291447.
```

T[n_] := Integrate[Sum[(-1)^(n-j+1) StirlingS2[n, j] j! x^j, {j,0,n}]^2, x];
Trow[n_] := CoefficientList[T[n], x] // Denominator;
Table[Trow[r], {r, 0, 7}] // Flatten

T(n,k) = Denominator([x^k] Integral(Sum_{j=0..n}(-1)^(n-j)*Stirling2(n,j)*j!*x^j)^m) for m = 2, n >= 0 and k = 0..m*n+1.

A290694 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, -1, 2, 0, 0, 1, -2, 3, 0, 0, -1, 14, -9, 24, 0, 0, 1, -10, 75, -48, 20, 0, 0, -1, 62, -135, 312, -300, 720, 0, 0, 1, -42, 903, -1680, 2800, -2160, 630, 0, 0, -1, 254, -1449, 40824, -21000, 27360, -17640, 4480

Offset: 0

Peter Luschny, Aug 24 2017

Consider a family of integrals I_m(n) = Integral_{x=0..1} P'(n, x)^m with P'(n,x) = Sum_{k=0..n}(-1)^(n-k)*Stirling2(n, k)*k!*x^k (see A278075 for the coefficients).
I_1(n) are the Bernoulli numbers A164555/A027642, I_2(n) are the Bernoulli median numbers A212196/A181131, I_3(n) are the numbers A291449/A291450.
The coefficients of the polynomials P_n(x)^m are for m = 1 A290694/A290695 and for m = 2 A291447/A291448.
Only omega(Clausen(n)) = A001221(A160014(n,1)) = A067513(n) coefficients are rational numbers if n is even. For odd n > 1 there are two rational coefficients.
Let C_k(n) = [x^k] P_n(x), k > 0 and n even. Conjecture: k is a prime factor of Clausen(n) <=> k = denominator(C_k(n)) <=> k does not divide Stirling2(n, k-1)*(k-1)!. (Note that by a comment in A019538 Stirling2(n, k-1)*(k-1)! is the number of chain topologies on an n-set having k open sets.)

			Triangle starts:
[0, 1]
[0, 0,  1]
[0, 0, -1,   2]
[0, 0,  1,  -2,    3]
[0, 0, -1,  14,   -9,  24]
[0, 0,  1, -10,   75, -48,   20]
[0, 0, -1,  62, -135, 312, -300, 720]
The first few polynomials are:
P_0(x) = x.
P_1(x) =  (1/2)*x^2.
P_2(x) = -(1/2)*x^2 + (2/3)*x^3.
P_3(x) =  (1/2)*x^2 - 2*x^3 + (3/2)*x^4.
P_4(x) = -(1/2)*x^2 + (14/3)*x^3 - 9*x^4 + (24/5)*x^5.
P_5(x) =  (1/2)*x^2 - 10*x^3 + (75/2)*x^4 - 48*x^5 + 20*x^6.
P_6(x) = -(1/2)*x^2 + (62/3)*x^3 - 135*x^4 + 312*x^5 - 300*x^6 + (720/7)*x^7.
Evaluated at x = 1 this gives an additive decomposition of the Bernoulli numbers:
B(0) =     1 =    1.
B(1) =   1/2 =  1/2.
B(2) =   1/6 = -1/2 +  2/3.
B(3) =     0 =  1/2 -    2 + 3/2.
B(4) = -1/30 = -1/2 + 14/3 -    9 + 24/5.
B(5) =     0 =  1/2 -   10 + 75/2 -   48 +  20.
B(6) =  1/42 = -1/2 + 62/3 -  135 +  312 - 300 + 720/7.

Peter Luschny, Illustrating A290694.

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

Cf. A160014, A278075.

BG_row := proc(m, n, frac, val) local F, g, v;
F := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n):
g := x -> int(F(n,x)^m, x):
`if`(val = "val", subs(x=1, g(x)), [seq(coeff(g(x),x,j), j=0..m*n+1)]):
`if`(frac = "num", numer(%), denom(%)) end:
seq(BG_row(1, n, "num", "val"), n=0..16);         # A164555
seq(BG_row(1, n, "den", "val"), n=0..16);         # A027642
seq(print(BG_row(1, n, "num", "poly")), n=0..12); # A290694 (this seq.)
seq(print(BG_row(1, n, "den", "poly")), n=0..12); # A290695
# Alternatively:
T_row := n -> numer(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 6 do T_row(n) od;

T[n_, k_] := If[k > 0, Numerator[StirlingS2[n, k - 1]*(k - 1)! / k], 0]; Table[T[n, k], {n, 0, 8}, {k, 0, n+1}] // Flatten

T(n, k) = Numerator(Stirling2(n, k - 1)*(k - 1)!/k) if k > 0 else 0; for n >= 0 and 0 <= k <= n+1.

A290695 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 5, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Aug 24 2017

See A290694 for comments.

			Triangle starts:
[1, 1]
[1, 1, 2]
[1, 1, 2, 3]
[1, 1, 2, 1, 2]
[1, 1, 2, 3, 1, 5]
[1, 1, 2, 1, 2, 1, 1]
[1, 1, 2, 3, 1, 1, 1, 7]
[1, 1, 2, 1, 2, 1, 1, 1, 1]

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

T_row := n -> denom(PolynomialTools:-CoefficientList(add((-1)^(n-j+1)*Stirling2(n, j-1)*(j-1)!*x^j/j, j=1..n+1), x)): for n from 0 to 7 do T_row(n) od;

T[n_] := Denominator[CoefficientList[Sum[(-1)^(n-j+1) StirlingS2[n, j-1] (j-1)! x^j/j, {j, 1, n+1}], x]];
Table[T[n], {n, 0, 7}] (* Jean-François Alcover, Jun 15 2019, from Maple *)

T(n, k) = Denominator([x^k] Integral (Sum_{j=0..n} (-1)^(n-j)*Stirling2(n,j)*j!* x^j)^m) for m = 1 and k = 0..n+1.

A291450 Denominators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n}(-1)^(n-k)* Stirling2(n, k)k!x^k.

Original entry on oeis.org

1, 4, 140, 28, 20020, 4004, 6466460, 184756, 148728580, 29745716, 133706993420, 2431036244, 449741705140, 31885268, 670910837521540, 134182167504308, 409926521725660940, 4822664961478364, 1278006214791766460, 1921813856829724, 242081282475556183660, 4401477863191930612

Offset: 0

Peter Luschny, Aug 24 2017

See A291449 and A290694 for comments.

Cf. A164555/A027642, A212196/A181131, A291449/A291450, A290694/A290695, A291447/A291448.

# Function BG_row is defined in A290694.
seq(BG_row(3, n, "den", "val"), n=0..20);

P[n_, x_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k!*x^k, {k, 0, n}];
a[n_] := Integrate[P[n, x]^3, {x, 0, 1}] // Denominator;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jun 15 2019 *)

cp's OEIS Frontend

A291447 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = BernoulliMedian(n).

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A291448 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n,x) such that Integral_{x=0..1} P'(n,x) = BernoulliMedian(n).

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A290694 Triangle read by rows, numerators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

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Maple

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A290695 Triangle read by rows, denominators of coefficients (in rising powers) of rational polynomials P(n, x) such that Integral_{x=0..1} P'(n, x) = Bernoulli(n, 1).

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Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A291450 Denominators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n}(-1)^(n-k)* Stirling2(n, k)*k!*x^k.

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Comments

Crossrefs

Programs

Maple

Mathematica

A291450 Denominators of Integral_{x=0..1} P(n, x)^3 with P(n, x) = Sum_{k=0..n}(-1)^(n-k)* Stirling2(n, k)k!x^k.