A053657 -id:A053657 - cp's OEIS frontend

A290761 Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

3, 5, -6, 16, 1, 7, 16, 28, 0, 15, 225, 1265, 3707, 7120, 4900, -6480, 27648, 3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0, 63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400, 9, 531, 14010, 219106

Offset: 1

Gregory Gerard Wojnar, Aug 09 2017

The polynomials come in pairs: first of odd degree; second of even degree 1 greater, whose constant term is always zero. Observations: All coefficients are positive except for the linear coefficients of the first polynomial in each pair, which are always negative. From the first of one pair to the first of the next pair, the degree always grows by 4. The "standard" factors of polynomials yielding the columns of triangle A290053 (beginning with column 3) are always of the form (1/A053657(k+2))*(N + k + 2) in odd rows of this triangle A290761, and of the form (N/A053657(k+2))*(N + k + 3)^2 in even rows of this triangle, where k is the row number. See examples.

			The first rows of the triangle are parsed as follows:
3, 5, -6, 16;
1, 7, 16, 28, 0;
15, 225, 1265, 3707, 7120, 4900, -6480, 27648;
3, 83, 961, 6201, 24708, 60700, 87968, 85056, 0;
63, 2457, 41580, 404866, 2532971, 10651177, 30102338, 56577724, 72856616, 36562176, -51101568, 298598400;
9, 531, 14010, 219106, 2266137, 16325259, 83797380, 307998768, 802828704, 1433652560, 1651979520, 1239918336, 0.
The associated full polynomials giving the columns of triangle A290053 are then:
(1/24) * (N + 3) * (3*N^3 + 5*N^2 - 6*N + 16);
(N/48) * (N + 5)^2 * (1*N^3 + 7*N^2 + 16*N + 28);
(1/5760) * (N + 5) * (15*N^7 + 225*N^6 + 1265*N^5 + 3707*N^4 + 7120*N^3 + 4900*N^2 - 6480*N + 27648);
(N/11520) * (N + 7)^2 * (3*N^7 + 83*N^6 + 961*N^5 + 6201*N^4 + 24708*N^3 + 60700*N^2 + 87968*N + 85056); etc.

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

The first column of this triangle is A290030; alternating entries of the first column give A260326. See also triangle A290053, whose columns are A000012-A000096, A290061-A290071, A290127-A290723, etc.

A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 96, 192, 1152, 768, 1536, 3072, 18432, 36864, 221184, 147456, 884736, 1769472, 10616832, 21233664, 637009920, 424673280, 2548039680, 5096079360, 152882380800, 61152952320, 366917713920, 81537269760, 163074539520, 326149079040, 1956894474240

Offset: 0

Peter Luschny, Feb 06 2021

nonn

The challenge is to characterize the sequence purely arithmetically, i.e., without reference to the Eulerian numbers or the Bernoulli polynomials.

András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences In: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (eds), Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA., 2006.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT] 2017, Amer. Math. Monthly.

Cf. A100655, A053657 (Minkowski), A341107, A341108, A318256, A144845, A163176, A201637 (Eulerian2), A036689, A324370, A007947, A324369, A195441.

Epoly := proc(n, x) add(combinat:-eulerian2(n, k)*binomial(x+k, 2*n), k = 0..n) / mul(j-x, j = 1..n): simplify(expand(%)) end:
seq(denom(Epoly(n, x)) / (n!*denom(bernoulli(n, x))), n = 0..30);

A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k,0,n}],{p, Prime[Range[n]]}];
A144845[n_] := Denominator[Together[BernoulliB[n, x]]];
A163176[n_] := A053657[n] / n!;
Table[(n + 1) A163176[n + 1] / A144845[n], {n, 0, 30}]

def A341109(n): # uses[A341108, A318256]
    return A341108(n)//A318256(n)
print([A341109(n) for n in (0..30)])

a(n) = A053657(n+1)/(n!*A144845(n)).
a(n) = (n+1)*A163176(n+1)/A144845(n).
a(n) = A341108(n)/A318256(n).
a(n) = A341107(n)*A324369(n+1).
a(n) = A341108(n)/A324370(n+1).
a(n) = A341108(n)*A007947(n+1)/A144845(n).
a(n) = A341108(n)*A324369(n+1)/A195441(n).
prime(n) divides a(k) for k >= A036689(n).
2^(n-1) divides exactly a(n) for n >= 2.

A202717 Triangle of numerators of coefficients of the polynomial Q^(3)m(n) defined by the recursion Q^(3)_0(n)=1; for m>=1, Q^(3)_m(n) = Sum{i=1...n} i^3*Q^(3)_(m-1)(i).

Original entry on oeis.org

1, 1, 2, 1, 0, 0, 21, 132, 294, 252, 21, -56, 0, 8, 0, 35, 450, 2293, 5700, 6405, 770, -3661, -240, 2320, 40, -672, 0, 0, 9555, 207480, 1889316, 9216312, 25051026, 33229560, 3678948, -35339304, -2666157, 51171120, 2178176, -49878192, -792064, 24460800, 4160, -3714816, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 23 2011

For m>=0, the denominator for all 4*m+1 terms of the m-th row is A202368(m+1).

See comment to A175669.

			The sequence of polynomials begins
Q^(3)_0=1,
Q^(3)_1=(x^4+2*x^3+x^2)/4,
Q^(3)_2=(21*x^8+132*x^7+294*x^6+252*x^5+21*x^4-56*x^3+8*x)/672,
Q^(3)_3=(35*x^12+450*x^11+2293*x^10+5700*x^9+6405*x^8+770*x^7-3661*x^6-240*x^5+2320*x^4+40x^3-672*x^2)/13440.

Cf. A202339, A053657, A202367, A202368, A202369, A175699.

Q^(3)_n(1)=1.

A342645 Triangle read by rows: T(n,k) gives n! times the coefficient of x^k in the polynomial that describes the number of permutations on x letters with major index n.

Original entry on oeis.org

1, -1, 1, -2, -1, 1, 0, -7, 0, 1, 0, -14, -13, 2, 1, 120, -46, -65, -15, 5, 1, 0, 516, -356, -165, -5, 9, 1, 5040, 1392, 266, -1421, -280, 28, 14, 1, 0, 46320, 3772, -5740, -3871, -280, 98, 20, 1, 0, 215280, 212724, -26272, -31437, -7791, 126, 222, 27, 1

Offset: 0

Peter Kagey, Mar 17 2021

This n-th row describes a polynomial that eventually agrees with the n-th column of A008302.

Conjecture: For each m, T(n,n-m) is a polynomial of degree 2m whose leading coefficient is abs(A290030(m)/A053657(m+1)).

			n\k |    0       1       2       3       4      5    6    7   8  9
----+--------------------------------------------------------------
  0 |    1;
  1 |   -1,      1;
  2 |   -2,     -1,      1;
  3 |    0,     -7,      0,      1;
  4 |    0,    -14,    -13,      2,      1;
  5 |  120,    -46,    -65,    -15,      5,     1;
  6 |    0,    516,   -356,   -165,     -5,     9,   1;
  7 | 5040,   1392,    266,  -1421,   -280,    28,  14,   1;
  8 |    0,  46320,   3772,  -5740,  -3871,  -280,  98,  20,  1;
  9 |    0, 215280, 212724, -26272, -31437, -7791, 126, 222, 27, 1;
For n = 4, the polynomial that describes the 4th column of A008302 is
A008302(x,4) = (-14x -13x^2 +2x^3 + x^4)/4! = Sum_{j=0..4} (T(j,4)*x^j)/4!.

Peter Kagey, Rows n = 0..100, flattened
Mike Earnest, Does the number of permutations in S_n with major index equal to k, satisfy a degree k polynomial?, Mathematics Stack Exchange answer.

Cf. A008302, A053657, A290030.

A008302T[0, 0] := 1; A008302T[-1, k_] := 0;
A008302T[n_, k_] := (A008302T[n, k] = If[0 <= k <= n*(n - 1)/2, A008302T[n, k - 1] + A008302T[n - 1, k] - A008302T[n - 1, k - n], 0]);
A342645Row[n_] := (A342645Row[n] = Expand[n!*InterpolatingPolynomial[Table[{m, A008302T[m, n]}, {m, n, 2*n + 2}], x]]);
A342645T[n_, k_] := Coefficient[A342645Row[n], x, k];

Conjectures:
T(n,n)   = 1.
T(n,n-1) = (-3n + n^2)/2.
T(n,n-2) = (-2n + 21n^2 - 22n^3 + 3n^4)/24.
T(n,n-3) = (96n - 134n^2 + 13n^3 + 37n^4 - 13n^5 + n^6)/48.

A153359 Scaled coefficients of the M. O. Rubinstein polynomials.

Original entry on oeis.org

1, -1, 1, -2, -1, 3, -2, -1, 2, 1, -152, -78, 125, 90, 15, -216, -114, 157, 135, 35, 3, -41424, -22444, 27552, 26551, 8505, 1197, 63, -66000, -36620, 40976, 42917, 15652, 2814, 252, 9, -13037952, -7390832, 7652084, 8557940, 3414775, 714840, 83790, 5220, 135, -21995904

Offset: 0

Peter Luschny, Dec 24 2008

The polynomials alpha_{k}(s) are defined in formula (1.4) in the paper cited below. The coefficients are in ascending order.

			alpha_{0}(t) = 1 / 1;
alpha_{1}(t) = (-1 + t) / 2;
alpha_{2}(t) = (-2 - t + 3t^2) / 24;
alpha_{3}(t) = (-2 - t + 2t^2 + t^3) / 48;

M. O. Rubinstein, Identities for the Riemann Zeta function., arXiv:0812.2592 [math.NT]

Cf. A053657.

alpha[0, ] = 1; alpha[k, s_] := (s - 1)/(k + 1) - Sum[((j - (s - 1)*(k - j))/(k - j + 1))*alpha[j, s]/(k), {j, 1, k - 1}] // Expand;
a53657[n_] := Product[p^Sum[Floor[(n - 1)/((p - 1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}];
row[k_] := CoefficientList[alpha[k, t]*a53657[k + 1], t];
Table[row[k], {k, 0, 7}] // Flatten (* Jean-François Alcover, Jul 19 2018 *)

The coefficients of the polynomials alpha_{k}(s)*A053657(k) where alpha_{0}(s) = 1 and alpha_{k+1}(s) = (s-1)/(k+2)-sum(j=1..k,((j-(s-1)*(k-j+1))/(k-j+2))*alpha_{j}(s))/(k+1).

More terms from Giovanni Resta, Jul 19 2018

A163394 The odd part of Minkowski(n)/n!

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 9, 9, 15, 3, 9, 3, 945, 135, 27, 27, 405, 45, 8505, 1701, 66825, 6075, 18225, 6075, 995085, 76545, 8505, 1215, 18225, 1215, 841995, 841995, 6506325, 382725, 32805, 3645, 850797675, 44778825, 3444525

Offset: 0

Peter Luschny, Jul 26 2009

nonn

Cf. A000265, A053657, A163176.

a := proc(n) local L,p;
L := proc(n,p,r) local q,s,m; m:=n-r; q:=p-r; s:=0;
do if q > n then break fi; s := s+iquo(m,q);
q := q*p od; s end; mul(p^(L(n,p,1)-L(n,p,0)),
p = select(isprime,[$3..n])); end

L[n_, p_, r_] := Module[{q, s, m}, m = n-r; q = p-r; s = 0; While[True, If[ q > n, Break[]]; s = s + Quotient[m, q]; q = q*p]; s];
a[n_] := Product[p^(L[n, p, 1]-L[n, p, 0]), {p, Select[Range[3, n], PrimeQ] }];
Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Jun 17 2019 *)

a(n) = A000265(A163176(n)). - Michel Marcus, Jun 17 2019

A202749 Triangle of numerators of coefficients of the polynomial Q^(4)m(n) defined by the recursion Q^(4)_0(n)=1; for m>=1,Q^(4)_m(n)=sum{i=1,...,n}i^4Q^(4)(m-1)(i). For m>=0, the denominator for all 5m+1 terms of the m-th row is A202369(m+1).

Original entry on oeis.org

1, 6, 15, 10, 0, -1, 0, 36, 280, 795, 900, 88, -450, -20, 200, 1, -30, 0, 19656, 311220, 1991430, 6354075, 9367722, 1283100, -10854935, -1064700, 16237338, 615615, -16336320, -136500, 8189909, 8190, -1243800, 0

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 23 2011

See comment in A175669.

			The sequence of polynomials begins
Q^(3)_0=1,
Q^(3)_1=(6*x^5+15*x^4+10*x^3-x)/30,
Q^(3)_2=(36*x^10+280*x^9+795*x^8+900*x^7+88*x^6-450*x^5-20*x^4+200*x^3+x^2-30*x)/1800.

Cf. A202339, A053657, A202367, A202368, A202369, A175699, A202717

Q^(4)_n(1)=1.

cp's OEIS Frontend

A290761 Irregular triangle, read by rows, of coefficients of polynomials that are the "nonstandard" factor of polynomials yielding the columns (up to sign) of triangle A290053, beginning with column 3.

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A341109 a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

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A202717 Triangle of numerators of coefficients of the polynomial Q^(3)m(n) defined by the recursion Q^(3)_0(n)=1; for m>=1, Q^(3)_m(n) = Sum{i=1...n} i^3*Q^(3)_(m-1)(i).

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A342645 Triangle read by rows: T(n,k) gives n! times the coefficient of x^k in the polynomial that describes the number of permutations on x letters with major index n.

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A153359 Scaled coefficients of the M. O. Rubinstein polynomials.

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A163394 The odd part of Minkowski(n)/n!

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A202749 Triangle of numerators of coefficients of the polynomial Q^(4)m(n) defined by the recursion Q^(4)_0(n)=1; for m>=1,Q^(4)_m(n)=sum{i=1,...,n}i^4*Q^(4)(m-1)(i). For m>=0, the denominator for all 5*m+1 terms of the m-th row is A202369(m+1).

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A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

A202749 Triangle of numerators of coefficients of the polynomial Q^(4)m(n) defined by the recursion Q^(4)_0(n)=1; for m>=1,Q^(4)_m(n)=sum{i=1,...,n}i^4Q^(4)(m-1)(i). For m>=0, the denominator for all 5m+1 terms of the m-th row is A202369(m+1).