A101686 -id:A101686 - cp's OEIS frontend

A375839 a(n) = Product_{k=0..n} (k^2 + n).

0, 2, 36, 1008, 41600, 2381400, 180457200, 17467670528, 2100621828096, 306960977700000, 53529274174376000, 10973787848179200000, 2611472797582941487104, 713649909809783275801472, 221870902844468552220000000, 77837994361783539267010560000

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A156648, A101686.

Cf. A375840, A375841, A375842, A375843, A375844, A375845.

Table[Product[k^2 + n, {k, 0, n}], {n, 0, 15}]
Round[Table[Sqrt[n] * Gamma[1 - I*Sqrt[n] + n] * Gamma[1 + I*Sqrt[n] + n] * Sinh[Sqrt[n]*Pi] / Pi, {n, 0, 15}]]

a(n) ~ n^(2*n + 3/2) / exp(2*n - Pi*n^(1/2) + 1).

A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969).

Original entry on oeis.org

1, -1, 1, 4, -5, 1, -36, 49, -14, 1, 576, -820, 273, -30, 1, -14400, 21076, -7645, 1023, -55, 1, 518400, -773136, 296296, -44473, 3003, -91, 1, -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1, 1625702400, -2483133696, 1017067024, -173721912, 14739153, -669188, 16422, -204, 1

Offset: 1

M. F. Hasler, Feb 03 2012

This is a signed version of A008955 with rows in reverse order. - Peter Luschny, Feb 04 2012

			Triangle starts:
  [1]         1;
  [2]        -1,        1;
  [3]         4,       -5,         1;
  [4]       -36,       49,       -14,       1;
  [5]       576,     -820,       273,     -30,       1;
  [6]    -14400,    21076,     -7645,    1023,     -55,    1;
  [7]    518400,  -773136,    296296,  -44473,    3003,  -91,    1;
  [8] -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1;

José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024. See p. 6.
Mark W. Coffey and Matthew C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

Cf. A036969, A008955, A008275, A121408, A001044 (column 1), A101686 (alternating row sums), A234324 (central terms).

# From Peter Luschny, Feb 29 2024: (Start)
ogf := n -> local j; z^2*mul(z^2 - j^2, j = 1..n-1):
Trow := n -> local k; seq(coeff(expand(ogf(n)), z, 2*k), k = 1..n):
# Alternative:
f := w -> (w^sqrt(t) + w^(-sqrt(t)))/2: egf := f((x/2 + sqrt(1 + (x/2)^2))^2):
ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, k), k = 1..n):  # (End)
# Assuming offset 0:
rowpoly := n -> (-1)^n * pochhammer(1 - sqrt(x), n) * pochhammer(1 + sqrt(x), n):
row := n -> local k; seq(coeff(expand(rowpoly(n)), x, k), k = 0..n):
seq(print(row(n)), n = 0..7);  # Peter Luschny, Aug 03 2024

rows = 10;
t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k - j)/((k - j)!*(k + j)!), {j, 1, k}];
T = Table[t[n, k], {n, 1, rows}, {k, 1, rows}] // Inverse;
Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018 *)

select(concat(Vec(matrix(10,10,n,k,T(n,k)/*from A036969*/)~^-1)), x->x)

def A204579(n, k): return (-1)^(n-k)*A008955(n, n-k)
for n in (0..7): print([A204579(n, k) for k in (0..n)]) # Peter Luschny, Feb 05 2012

T(n, k) = (-1)^(n-k)*A008955(n, n-k). - Peter Luschny, Feb 05 2012
T(n, k) = Sum_{i=k-n..n-k} (-1)^(n-k+i)*s(n,k+i)*s(n,k-i) = Sum_{i=0..2*k} (-1)^(n+i)*s(n,i)*s(n,2*k-i), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 07 2012
From Peter Bala, Aug 29 2012: (Start)
T(n, k) = T(n-1, k-1) - (n-1)^2*T(n-1, k). (Recurrence equation.)
Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/{(2*n)!/2^n} and
L(x) = 2*{arcsinh(sqrt(x/2))}^2 = Sum_{n >=1} (-1)^n*(n-1)!^2*x^n/{(2*n)!/2^n}.
L(x) is the compositional inverse of E(x) - 1.
A generating function for the triangle is E(t*L(x)) = 1 + t*x + t*(-1 + t)*x^2/6 + t*(4 - 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1,1,6,90,...] is given by (2*n)!/2^n. Cf. A008275 with generating function exp(t*log(1+x)).
The e.g.f. is E(t*L(x^2/2)) = cosh(2*sqrt(t)*arcsinh(x/2)) = 1 + t*x^2/2! + t*(t-1)*x^4/4! + t*(t-1)*(t-4)*x^6/6! + .... (End)
From Peter Luschny, Feb 29 2024: (Start)
T(n, k) = [z^(2*k)] z^2*Product_{j=1..n-1} (z^2 - j^2).
T(n, k) = (2*n)! * [t^k] [x^(2*n)] (w^sqrt(t) + w^(-sqrt(t)))/2 where w = (x/2 + sqrt(1 + (x/2)^2))^2. (End)
T(n, k) = [x^k] (-1)^n * Pochhammer(1 - sqrt(x), n) * Pochhammer(1 + sqrt(x), n), assuming offset 0. - Peter Luschny, Aug 03 2024
Integral_{0..oo} x^s / (cosh(x))^(2*n) dx = (2^(2*n - s - 1) * s! * (-1)^(n-1)) / (2*n - 1)!)*Sum_{k=1..n} T(n,k)*DirichletEta(s - 2*k + 2). - Ammar Khatab, Apr 11 2025

Typo in data corrected by Peter Luschny, Feb 05 2012

A293290 a(n) = Product_{1 <= j <= k <= n} (k^2 + j^2).

Original entry on oeis.org

1, 2, 80, 187200, 50918400000, 2675955409920000000, 40702283662588674048000000000, 250658664786823821917343252480000000000000, 832906513114759565863066815448211678822400000000000000000, 1919381816160714520414106848157314737202346840876384256000000000000000000000

Offset: 0

Velin Yanev, Oct 05 2017

Suggested by Omar E. Pol from A264596 formula.

Cf. A101686, A203475, A324403.

Table[Product[k^2 + j^2, {k, 1, n}, {j, 1, k}], {n, 0, 10}]

[prod([prod([k^2+j^2 for j in range(1,k+1)]) for k in range(1,n+1)]) for n in range(10)] # Danny Rorabaugh, Oct 16 2017

a(n) ~ sqrt(Gamma(1/4)) * Pi^(-1/8) * 2^(n^2/2 + n - 1/8) * exp(Pi*n*(n+1)/4 - 3*n^2/2 - n + Pi/24) * n^(n*(n+1) + 1/4). - Vaclav Kotesovec, Feb 26 2019

A051893 a(n) = Sum_{i=1..n-1} i^2*a(i), a(1) = 1.

Original entry on oeis.org

1, 1, 5, 50, 850, 22100, 817700, 40885000, 2657525000, 217917050000, 22009622050000, 2685173890100000, 389350214064500000, 66189536390965000000, 13039338669020105000000, 2946890539198543730000000, 757350868574025738610000000, 219631751886467464196900000000

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 17 1999

Cf. A001710, A101686, A256019, A256020.

a := n -> `if`(n=1,1,(sinh(Pi)*GAMMA(n-I)*GAMMA(n+I))/(2*Pi)):
seq(simplify(a(n)), n=1..18); # Peter Luschny, Oct 19 2016

a[n_] := Pochhammer[2-I, n-2]*Pochhammer[2+I, n-2]; a[1] = 1; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 21 2012, after Vladeta Jovovic *)
Join[{1},FoldList[Times,1,Range[2,20]^2+1]] (* Harvey P. Dale, Jul 04 2013 *)
Clear[a]; a[1]=1; a[n_]:=a[n]=Sum[i^2*a[i],{i,1,n-1}]; Table[a[n],{n,1,20}] (* Vaclav Kotesovec, Mar 13 2015 *)

a(n) = Product_{i=2..n-1} (i^2+1), for n>2. - Vladeta Jovovic, Nov 26 2002
From Vaclav Kotesovec, Mar 13 2015: (Start)
For n > 1, a(n) = A101686(n-1)/2.
a(n) ~ (n-1)!^2 * sinh(Pi)/(2*Pi).
(End)
a(n) = (A003703(n)^2 + A009454(n)^2 + A000007(n-1))/2. - Vladimir Reshetnikov, Oct 15 2016
a(n) = sinh(Pi)*Gamma(n-I)*Gamma(n+I)/(2*Pi) for n>1. - Peter Luschny, Oct 19 2016

More terms from Harvey P. Dale, Jul 04 2013

A242652 Imaginary part of Product_{k=0..n} (i-k), where i=sqrt(-1).

Original entry on oeis.org

1, -1, 1, 0, -10, 90, -730, 6160, -55900, 549900, -5864300, 67610400, -839594600, 11186357000, -159300557000, 2416003824000, -38894192662000, 662595375078000, -11911522255750000, 225382826562400000, -4477959179352100000, 93217812901913700000, -2029107997508660900000, 46099220630461596000000

Offset: 0

N. J. A. Sloane, May 29 2014

sign

Shifted version of A009454. - R. J. Mathar, May 30 2014

			Table of n, Product_{k=0..n} (i-k):
   0, i
   1, -1 - i
   2, 3 + i
   3, -10
   4, 40 - 10*i
   5, -190 + 90*i
   6, 1050 - 730*i
   7, -6620 + 6160*i
   8, 46800 - 55900*i
   9, -365300 + 549900*i
  10, 3103100 - 5864300*i
  11, -28269800 + 67610400*i
  12, 271627200 - 839594600*i

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 561-562, Section 148.

Cf. A009454.
Cf. A242651, A101686.
A231531 is the same except for signs.

a:= n-> Im(mul(I-j, j=0..n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 03 2021

a(n) = imag(prod(k=0, n, I-k)); \\ Michel Marcus, Jan 03 2021

A242651 Real part of Product_{k=0..n} (i-k), where i = sqrt(-1).

Original entry on oeis.org

0, -1, 3, -10, 40, -190, 1050, -6620, 46800, -365300, 3103100, -28269800, 271627200, -2691559000, 26495469000, -238131478000, 1394099824000, 15194495654000, -936096296850000, 29697351895900000, -819329864480400000, 21683886333440500000, -570263312237604700000, 15145164178973569000000

Offset: 0

N. J. A. Sloane, May 29 2014

sign

Shifted version of A003703. - R. J. Mathar, May 30 2014

			Table of n, Product_{k=0..n} (i-k):
   0,         i
   1,        -1 -           i
   2,         3 +           i
   3,       -10
   4,        40 -        10*i
   5,      -190 +        90*i
   6,      1050 -       730*i
   7,     -6620 +      6160*i
   8,     46800 -     55900*i
   9,   -365300 +    549900*i
  10,   3103100 -   5864300*i
  11, -28269800 +  67610400*i
  12, 271627200 - 839594600*i

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, pp. 561-562, Section 148.

Cf. A003703, A242652, A101686.

A231531 is the same except for signs.

Table[Re[(I - n)*Pochhammer[1 + I - n, n]], {n, 0, 25}] (* Vaclav Kotesovec, May 23 2021 *)

a(n) = real(prod(k=0, n, I-k)); \\ Michel Marcus, Jan 03 2021

a(n) = Sum_{k=0..floor((n+1)/2)} (-1)^k*Stirling1(n+1,2*k). - Ammar Khatab, May 23 2021

A277354 a(n) = Product_{k=1..n} (4*k^2+1).

Original entry on oeis.org

1, 5, 85, 3145, 204425, 20646925, 2993804125, 589779412625, 151573309044625, 49261325439503125, 19753791501240753125, 9580588878101765265625, 5527999782664718558265625, 3742455852864014463945828125, 2937827844498251354197475078125

Offset: 0

Vaclav Kotesovec, Oct 10 2016

nonn

In general, for m>0, Product_{k=1..n} (m*k^2+1) is asymptotic to 2*m^(n+1/2) * n^(2*n+1) * sinh(Pi/sqrt(m)) / exp(2*n).

Cf. A079484, A101686, A101928, A274306, A277352, A277353.

Table[Product[4*k^2+1, {k, 1, n}], {n, 0, 15}]
Round@Table[2^(2 n + 1) Abs[Gamma[1 + I/2 + n]]^2 Sinh[Pi/2]/Pi, {n, 0, 15}] (* Vladimir Reshetnikov, Oct 10 2016 *)

a(n) = prod(k=1, n, (4*k^2+1)); \\ Michel Marcus, Oct 11 2016

a(n) = (-1)^(n+1) * A101928(n+2).
a(n) ~ 2^(2*n+2) * n^(2*n+1) * sinh(Pi/2) / exp(2*n).
a(n) = 2^(2*n+1) * |Gamma(1 + i/2 + n)|^2 * sinh(Pi/2)/Pi. - Vladimir Reshetnikov, Oct 10 2016

A375041 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+1. See Comments.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 17, 10, 1, 18, 97, 180, 100, 1, 35, 403, 1829, 3160, 1700, 1, 61, 1313, 12307, 50714, 83860, 44200, 1, 98, 3570, 60888, 506073, 1960278, 3147020, 1635400, 1, 148, 8470, 239388, 3550473, 27263928, 101160920, 158986400, 81770000, 1, 213

Offset: 1

Clark Kimberling, Sep 11 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
  1 + x,
  1 + 3 x + 2 x^2,
  1 + 8 x + 17 x^2 + 10 x^3.
First 5 rows of array:
  1    1
  1    3     2
  1    8    17    10
  1   18    97   180   100
  1   35  4034  1829  3160  1700

Cf. A000290, A081489 (column 2), A101686 (T(n,n+1)), A374848, A375042, A375043.

s[n_] := n^2  x; t[n_] := 1 + x;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A375043 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+2. See Comments.

Original entry on oeis.org

2, 1, 4, 6, 2, 8, 32, 34, 10, 16, 144, 388, 360, 100, 32, 560, 3224, 7316, 6320, 1700, 64, 1952, 21008, 98456, 202856, 167720, 44200, 128, 6272, 114240, 974208, 4048584, 7841112, 6294040, 1635400, 256, 18944, 542080, 7660416, 56807568, 218111424, 404643680

Offset: 1

Clark Kimberling, Sep 14 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
2 + x,
4 + 6 x + 2 x^2,
8 + 32 x + 34 x^2 + 10 x^3.
First 5 rows of array:
 2    1
 4    6     2
 8   32    34    10
16  144   388   360   100
32  560  3224  7316  6320  1700

Cf. A000290, A101686 (T(n,n+1)), A374848, A375041, A375042.

s[n_] := n^2  x; t[n_] := x + 2;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A263766 a(n) = Product_{k=1..n} (k^2 - 2).

Original entry on oeis.org

1, -1, -2, -14, -196, -4508, -153272, -7203784, -446634608, -35284134032, -3457845135136, -411483571081184, -58430667093528128, -9757921404619197376, -1893036752496124290944, -422147195806635716880512, -107225387734885472087650048

Offset: 0

Vladimir Reshetnikov, Oct 25 2015

			For n = 3, a(3) = (1^2 - 2)*(2^2 - 2)*(3^2 - 2) = -14.
G.f. = 1 - x - 2*x^2 - 14*x^3 - 196*x^4 - 4508*x^5 - 153272*x^6 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A101686, A263688, A263687.

Cf. A008865.

a263766 n = a263766_list !! n
a263766_list = scanl (*) 1 a008865_list
-- Reinhard Zumkeller, Oct 26 2015

Table[Product[k^2 - 2, {k, 1, n}], {n, 0, 16}]
Expand@Table[-Pochhammer[Sqrt[2], n+1] Pochhammer[-Sqrt[2], n+1]/2, {n, 0, 16}]
Join[{1},FoldList[Times,Range[20]^2-2]] (* Harvey P. Dale, Aug 14 2022 *)

a(n) = prod(k=1, n, k^2-2); \\ Michel Marcus, Oct 25 2015

a(n) = Gamma(1+sqrt(2)+n)*Gamma(1-sqrt(2)+n)*sin(Pi*sqrt(2))/(Pi*sqrt(2)).
a(n) = A263688(n+1)^2-A263687(n+1)^2/2.
a(n) ~ exp(-2*n)*n^(2*n+1)*sqrt(2)*sin(Pi*sqrt(2)).
G.f. for 1/a(n): hypergeom([1],[1-sqrt(2),1+sqrt(2)], x).
E.g.f. for 1/a(n): hypergeom([],[1-sqrt(2),1+sqrt(2)], x).
E.g.f. for a(n)/n!: hypergeom([1-sqrt(2),1+sqrt(2)], [1], x).
Recurrence: a(0) = 1, a(n) = (n^2-2)*a(n-1).
0 = a(n)*(-24*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) if n>=0. - Michael Somos, Oct 30 2015

cp's OEIS Frontend

A375839 a(n) = Product_{k=0..n} (k^2 + n).

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A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2*n, 2*k) (A036969).

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A293290 a(n) = Product_{1 <= j <= k <= n} (k^2 + j^2).

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A051893 a(n) = Sum_{i=1..n-1} i^2*a(i), a(1) = 1.

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A242652 Imaginary part of Product_{k=0..n} (i-k), where i=sqrt(-1).

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A242651 Real part of Product_{k=0..n} (i-k), where i = sqrt(-1).

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A277354 a(n) = Product_{k=1..n} (4*k^2+1).

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A375041 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+1. See Comments.

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A204579 Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969).