A248668 -id:A248668 - 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 *)

A343832 a(n) = Sum_{k=0..n} k! * binomial(n,k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 4, 31, 358, 5509, 106096, 2456299, 66471826, 2059640713, 71920704124, 2794938616471, 119653108240414, 5595650767265101, 283841520215780008, 15523069639558351459, 910529206043204428426, 57023540590242398853649, 3797750659849704886903156, 268025698704886063968108943

Offset: 0

Seiichi Manyama, May 01 2021

nonn

Let A(x) be the e.g.f. of this sequence, and B(x) be the e.g.f. of A082545, then A(x)/B(x) = C(x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108). This follows from the fact that this sequence and A082545 form adjacent semi-diagonals of table A088699. - Paul D. Hanna, Aug 16 2022

Seiichi Manyama, Table of n, a(n) for n = 0..365
Wikipedia, Laguerre polynomials
Index entries for sequences related to Laguerre polynomials

Cf. A000522, A045721, A082545, A152059, A248668, A343830, A343831, A000108.

[Factorial(n)*Evaluate(LaguerrePolynomial(n, n+1), -1): n in [0..40]]; // G. C. Greubel, Aug 11 2022

a := n -> add(k!*binomial(n, k)*binomial(2*n+1, k), k=0..n):
a := n -> n!*add(binomial(2*n+1, k)/(n-k)!, k=0..n):
a := n -> (-1)^n*KummerU(-n, n+2, -1):
a := n -> n!*LaguerreL(n, n+1, -1): # Peter Luschny, May 02 2021

a[n_] := Sum[k! * Binomial[n, k] * Binomial[2*n+1, k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, May 01 2021 *)
Table[(-1)^n * HypergeometricU[-n, 2 + n, -1], {n, 0, 20}] (* Vaclav Kotesovec, May 02 2021 *)

a(n) = sum(k=0, n, k!*binomial(n, k)*binomial(2*n+1, k));

a(n) = (2*n+1)!*sum(k=0, n, binomial(n, k)/(k+n+1)!);

a(n) = n!*sum(k=0, n, binomial(2*n+1, k)/(n-k)!);

PARI
```
a(n) = n!*pollaguerre(n, n+1, -1);
```

[factorial(n)*gen_laguerre(n, n+1, -1) for n in (0..40)] # G. C. Greubel, Aug 11 2022

a(n) = (2*n+1)! * Sum_{k=0..n} binomial(n,k)/(k+n+1)!.
a(n) = n! * Sum_{k=0..n} binomial(2*n+1,k)/(n-k)!.
a(n) = n! * LaguerreL(n, n+1, -1).
a(n) = n! * [x^n] exp(x/(1 - x))/(1 - x)^(n+2).
a(n) == 1 (mod 3).
a(n) ~ 2^(2*n + 3/2) * n^n / exp(n-1). - Vaclav Kotesovec, May 02 2021
From Paul D. Hanna, Aug 16 2022: (Start)
E.g.f.: exp( (1-2*x - sqrt(1-4*x))/(2*x) ) * (1 - sqrt(1-4*x)) / (2*x*sqrt(1-4*x)), derived from the e.g.f for A082545 given by Mark van Hoeij.
E.g.f.: exp(C(x) - 1) * C(x) / sqrt(1-4*x), where C(x) = (1 - sqrt(1-4*x))/(2*x) is the Catalan function (A000108). (End)

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.

A248665 Triangular array of coefficients of polynomials p(n,x) defined in Comments; these are the polynomials defined at A248664, but here the coefficients are written in the order of decreasing powers of x.

Original entry on oeis.org

1, 2, 2, 9, 12, 5, 64, 112, 68, 16, 625, 1375, 1125, 420, 65, 7776, 20736, 21600, 11124, 2910, 326, 117649, 369754, 470596, 311787, 114611, 22652, 1957, 2097152, 7602176, 11468800, 9342976, 4455424, 1254976, 196872, 13700, 43046721, 176969853, 309298662

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 (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)
First six rows of the triangle:
1
2     2
9     12     5
64    112    68     16
625   1375   1125   420    65
7776  20736  21600  11124  2910   326

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

Cf. A248664, 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] = Reverse[CoefficientList[p[x, n], x]];
TableForm[Table[c[n], {n, 1, 10}]] (* A248665 array *)
Flatten[Table[c[n], {n, 1, 10}]]   (* A248665 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*)

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.

cp's OEIS Frontend

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

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A343832 a(n) = Sum_{k=0..n} k! * binomial(n,k) * binomial(2*n+1,k).

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

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A248665 Triangular array of coefficients of polynomials p(n,x) defined in Comments; these are the polynomials defined at A248664, but here the coefficients are written in the order of decreasing powers of x.

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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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