A248665 -id:A248665 - cp's OEIS frontend

A248664 Triangular array of coefficients of polynomials p(n,k) defined in Comments.

1, 2, 2, 5, 12, 9, 16, 68, 112, 64, 65, 420, 1125, 1375, 625, 326, 2910, 11124, 21600, 20736, 7776, 1957, 22652, 114611, 311787, 470596, 369754, 117649, 13700, 196872, 1254976, 4455424, 9342976, 11468800, 7602176, 2097152, 109601, 1895148, 14699961, 65045025

Offset: 1

Clark Kimberling, Oct 11 2014

The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1).

These polynomials occur in connection with factorials of numbers of the form [n/k] = floor(n/k); e.g., Sum_{n >= 0} ([n/k]!^k)/n! = Sum_{n >= 0} (n!^k)*p(k,n)/(k*n + k - 1)!.

			The first six polynomials:
p(1,x) = 1
p(2,x) = 2 (1 + x)
p(3,x) = 5 + 12 x + 9x^2
p(4,x) = 4 (4 + 17 x + 28 x^2 + 16 x^3)
p(5,x) = 5 (13 + 84 x + 225 x^2 + 275 x^3 + 125 x^4)
p(6,x) = 2 (163 + 1455 x + 5562 x^2 + 10800 x^3 + 10368 x^4 + 3888 x^5)
First six rows of the triangle:
1
2     2
5     12     9
16    68    112    64
65    420   1125   1375    625
326   2910  11124  21600   20736   7776

Clark Kimberling, Table of n, a(n) for n = 1..5000

Cf. A248665, A248666, A248667, A248668, A248669.

t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]]  (* A248664 array *)
Flatten[Table[c[n], {n, 1, 10}]] (* A248664 sequence *)
u = Table[Apply[GCD, c[n]], {n, 1, 60}] (* A248666 *)
Flatten[Position[u, 1]]  (* A248667 *)
Table[Apply[Plus, c[n]], {n, 1, 60}]    (* A248668 *)
Table[p[x, n] /. x -> -1, {n, 1, 30}] (* A153229 signed *)

A248669 Triangular array of coefficients of polynomials q(n,k) defined in Comments.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 16, 17, 7, 1, 65, 84, 45, 11, 1, 326, 485, 309, 100, 16, 1, 1957, 3236, 2339, 909, 196, 22, 1, 13700, 24609, 19609, 8702, 2281, 350, 29, 1, 109601, 210572, 181481, 89225, 26950, 5081, 582, 37, 1, 986410, 2004749, 1843901, 984506, 331775

Offset: 1

Clark Kimberling, Oct 11 2014

q(n,x) = 1 + k+x + (k+x)(k-1+x) + (k+x)(k-1+x)(k-2+x) + ... + (k+x)(k-1+x)(k-2+x)...(1+x). The arrays at A248229 and A248664 have the same first column, given by A000522(n) for n >= 0. The alternating row sums of the array at A248669 are also given by A000522; viz., q(n,-1) = q(n-1,0) = A000522(n-2) for n >= 2. Column 2 of A248669 is given by A093344(n) for n >= 1.

			The first six polynomials:
p(1,x) = 1
p(2,x) = 2 + x
p(3,x) = 5 + 4 x + x^2
p(4,x) = 16 + 17 x + 7 x^2 + x^3
p(5,x) = 65 + 8 x + 45 x^2 + 11 x^3 + x^4
p(6,x) = 326 + 485 x + 309 x^2 + 100 x^3 + 16 x^4 + x^5
First six rows of the triangle:
1
2     1
5     4     1
16    17    7    1
65    84    45   11    1
326   485  309   100   16   1

Clark Kimberling, Table of n, a(n) for n = 1..5000

Cf. A248665, A248666, A248667, A248668, A248670.

t[x_, n_, k_] := t[x, n, k] = Product[x + n - i, {i, 1, k}];
q[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[q[x, n], {n, 1, 6}]];
TableForm[Table[Factor[q[x, n]], {n, 1, 6}]];
c[n_] := c[n] = CoefficientList[q[x, n], x];
TableForm[Table[c[n], {n, 1, 12}]] (* A248669 array *)
Flatten[Table[c[n], {n, 1, 12}]]   (* A248669 sequence *)

q(n,x) = (x + n - 1)*q(n-1,x) + 1, with q(1,x) = 1.

A248668 Sum of the numbers in the n-th row of the array at A248664.

Original entry on oeis.org

1, 4, 26, 260, 3610, 64472, 1409006, 36432076, 1087911890, 36844580000, 1395429571222, 58439837713556, 2681526361893626, 133783187672365480, 7210345924097089790, 417482356526745344732, 25844171201928905477026, 1703359919973405018460976

Offset: 1

Clark Kimberling, Oct 11 2014

The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1). For more, see A248664.

			The first six polynomials:
p(1,x) = 1
p(2,x) = 2 (x + 1)
p(3,x) = 9x^2 + 12 x +  5
p(4,x) = 4 (16 x^3 + 28 x^2 + 17 x + 4)
p(5,x) = 5 (125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
p(6,x) = 2 (3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that
a(1) = p(1,1) = 1, a(2) = p(2,1) = 4, a(3) = p(3,1) = 26.

Seiichi Manyama, Table of n, a(n) for n = 1..366

Cf. A000522, A248664, A248665, A248666, A248669, A294039.

with (combinat):
seq(add( k!*binomial(2*n-1,k),k = 0..n-1 ), n = 0..20);
# Peter Bala, Nov 14 2017

t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)
Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)

a(n) = sum(k = 0, n-1, k!*binomial(2*n-1,k)); \\ Michel Marcus, Nov 15 2017

a(n) = p(n,1), where p(n,x) is defined at A248664.
a(n) = Sum_{k = 0..n-1} k!*binomial(2*n-1,k). - Peter Bala, Nov 14 2017
a(n) = A294039(n) - Pochhammer(n, n)*A000522(n). - Peter Luschny, Nov 14 2017

A248666 Greatest common divisor of the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 4, 1, 10, 1, 4, 13, 2, 5, 4, 1, 2, 1, 20, 1, 2, 1, 4, 5, 26, 1, 4, 1, 10, 1, 4, 1, 2, 5, 4, 37, 2, 13, 20, 1, 2, 1, 4, 5, 2, 1, 4, 1, 10, 1, 52, 1, 2, 5, 4, 1, 2, 1, 20, 1, 2, 1, 4, 65, 2, 1, 4, 1, 10, 1, 4, 1, 74, 5, 4, 1, 26, 1, 20, 1

Offset: 1

Clark Kimberling, Oct 11 2014

The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1). For more, see A248664. For n such that the coefficients of p(n,x) are relatively prime, see A248667.

			The first six polynomials are shown here.  The number just to the right of "=" is the GCD of the coefficients.
p(1,x) = 1*1
p(2,x) = 2*(x + 1)
p(3,x) = 1*(9x^2 + 12 x +  5)
p(4,x) = 4*(16 x^3 + 28 x^2 + 17 x + 4)
p(5,x) = 5*(125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
p(6,x) = 2*(3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that A248666 = (1,2,1,4,5,2, ...).

Cf. A248664, A248665, A248667, A248668, A248669.

t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]]   (* A248664 array *)
Table[Apply[GCD, c[n]], {n, 1, 60}]  (* A248666 *)

A248667 Numbers k for which coefficients of the polynomial p(k,x) defined in Comments are relatively prime.

Original entry on oeis.org

1, 3, 7, 9, 11, 17, 19, 21, 23, 27, 29, 31, 33, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 93, 97, 99, 101, 103, 107, 109, 113, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141, 147, 149, 151, 153, 157, 159, 161, 163, 167

Offset: 1

Clark Kimberling, Oct 11 2014

The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1). For more, see A248664.
Since p(n,x) is a sum of products of terms (n*x + i), the only coefficient which is not necessarily divisible by n is the coefficient of x^0 = A000522(n-1). On the other hand, the coefficient of x^(n-1) is n^n. Therefore n is in this sequence iff gcd(n, A000522(n-1)) = 1. - Peter J. Taylor, Apr 08 2022
From Mikhail Kurkov, Apr 09 2022: (Start)
False conjecture (which still gives many correct values): {b(n)} is a subsequence of {a(n)} where {b(n)} are the numbers m for which Sum(abs(Moebius(p_j+1))) = 0 with m = Product(p_j^k_j). This conjecture was disproved by Peter J. Taylor. The first counterexample, i.e., the smallest m which belongs to {b(n)} and does not belong to {a(n)}, is m = 463. All other counterexamples computed up to 2.5*10^4 have the form 463*b(n). Are there any other numbers q such that q and q*b(n) are counterexamples for any n > 0? [verification needed]
Conjecture: any composite a(n) can be represented as a product a(i)*a(j) (i > 1, j > 1) in at least one way. (End)

			The first six polynomials with GCD(coefficients) shown just to the right of "=":
p(1,x) = 1
p(2,x) = 2*(x + 1)
p(3,x) = 1*(9x^2 + 12 x +  5)
p(4,x) = 4*(16 x^3 + 28 x^2 + 17 x + 4)
p(5,x) = 5*(125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
p(6,x) = 2*(3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that a(1) = 1 and a(2) = 3.

Cf. A000522, A248664, A248665, A248666, A248668, A248669.

t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)
u = Table[Apply[GCD, c[n]], {n, 1, 60}]  (* A248666 *)
Flatten[Position[u, 1]]  (* this sequence *)
Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)

A248670 Triangular array of coefficients of polynomials q defined in Comments; the coefficients are written in the order of decreasing powers of x.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 1, 7, 17, 16, 1, 11, 45, 84, 65, 1, 16, 100, 309, 485, 326, 1, 22, 196, 909, 2339, 3236, 1957, 1, 29, 350, 2281, 8702, 19609, 24609, 13700, 1, 37, 582, 5081, 26950, 89225, 181481, 210572, 109601, 1, 46, 915, 10319, 72679, 331775, 984506

Offset: 1

Clark Kimberling, Oct 11 2014

q(n,x) = 1 + k+x + (k+x)(k-1+x) + (k+x)(k-1+x)(k-2+x) + ... + (k+x)(k-1+x)(k-2+x)...(1+x). (See A248669.)

			The first six polynomials:
q(1,x) = 1
q(2,x) = x + 2
q(3,x) = x^2 + 4 x + 5
q(4,x) = x^3 + 7 x^2 + 17 x + 16
q(5,x) = x^4 + 11 x^3 + 45 x^2 + 8 x + 65
q(6,x) = x^5 + 16 x^4 + 100 x^3 + 309 x^2 + 485 x + 326
First six rows of the triangle:
1
1   2
1   4    5
1   7    17   16
1   11   45   84   65
1   16   100  309  485  326

Clark Kimberling, Table of n, a(n) for n = 1..5000

Cf. A248665, A248666, A248667, A248668, A248669.

t[x_, n_, k_] := t[x, n, k] = Product[x + n - i, {i, 1, k}];
q[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[q[x, n], {n, 1, 6}]];
TableForm[Table[Factor[q[x, n]], {n, 1, 6}]];
c[n_] := c[n] = Reverse[CoefficientList[q[x, n], x]];
TableForm[Table[c[n], {n, 1, 12}]] (* A248669 array *)
Flatten[Table[c[n], {n, 1, 12}]]   (* A248669 sequence *)

q(n,x) = (x + n - 1)*q(n-1,x) + 1, with q(1,x) = 1.

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A248664 Triangular array of coefficients of polynomials p(n,k) defined in Comments.

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A248669 Triangular array of coefficients of polynomials q(n,k) defined in Comments.

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A248668 Sum of the numbers in the n-th row of the array at A248664.

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A248666 Greatest common divisor of the coefficients of the polynomial p(n,x) defined in Comments.

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A248667 Numbers k for which coefficients of the polynomial p(k,x) defined in Comments are relatively prime.

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A248670 Triangular array of coefficients of polynomials q defined in Comments; the coefficients are written in the order of decreasing powers of x.

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