A080663 -id:A080663 - cp's OEIS frontend

A165674 Triangle generated by the asymptotic expansions of the E(x,m=2,n).

1, 3, 1, 11, 5, 1, 50, 26, 7, 1, 274, 154, 47, 9, 1, 1764, 1044, 342, 74, 11, 1, 13068, 8028, 2754, 638, 107, 13, 1, 109584, 69264, 24552, 5944, 1066, 146, 15, 1, 1026576, 663696, 241128, 60216, 11274, 1650, 191, 17, 1

Offset: 1

Johannes W. Meijer, Oct 05 2009

The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the E(x,m=2,n) ~ (exp(-x)/x^2)*(1 - (1+2*n)/x + (2+6*n+3*n^2)/x^2 - (6+22*n+18*n^2+ 4*n^3)/x^3 + ... ) is discussed in A028421. The formula for the asymptotic expansion leads for n = 1, 2, 3, .., to the left hand columns of the triangle given above.
The recurrence relations of the right hand columns of this triangle lead to Pascal's triangle A007318, their a(n) formulas lead to Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; cf. A080663, A165676, A165677, A165678 and A165679.
The row sums of this triangle lead to A093344. Surprisingly the e.g.f. of the row sums Egf(x) = (exp(1)*Ei(1,1-x) - exp(1)*Ei(1,1))/(1-x) leads to the exponential integrals in view of the fact that E(x,m=1,n=1) = Ei(n=1,x). We point out that exp(1)*Ei(1,1) = A073003.
The Maple programs generate the coefficients of the triangle given above. The first one makes use of a relation between the triangle coefficients, see the formulas, and the second one makes use of the asymptotic expansions of the E(x,m=2,n).
Amarnath Murthy discovered triangle A093905 which is the reversal of our triangle.
A165675 is an extended version of this triangle. Its reversal is A105954.
Triangle A094587 is generated by the asymptotic expansions of E(x,m=1,n).

A093905 is the reversal of this triangle.
A000254, A001705, A001711, A001716, A001721, A051524, A051545, A051560, A051562, A051564 are the first ten left hand columns.
A080663, n>=2, is the third right hand column.
A165676, A165677, A165678 and A165679 are the next right hand columns, A093344 gives the row sums.
A073003 is Gompertz's constant.
A094587 is generated by the asymptotic expansions of E(x, m=1, n).
Cf. A165675, A105954 (Quet) and A067176 (Bottomley).
Cf. A007318 (Pascal), A028421 (Wiggen), A126671 (Wood).

nmax:=9; for n from 1 to nmax do a(n, n) := 1 od: for n from 2 to nmax do a(n, 1) := n*a(n-1, 1) + (n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do a(n, m) := (n-m+1)*a(n-1, m) + a(n-1, m-1) od: od: seq(seq(a(n, m), m = 1..n), n = 1..nmax);
# End program 1
nmax := nmax+1: m:=2; with(combinat): EA := proc(x, m, n) local E, i; E:=0: for i from m-1 to nmax+2 do E := E + sum((-1)^(m+k1+1) * binomial(k1, m-1) * n^(k1-m+1) * stirling1(i, k1), k1=m-1..i) / x^(i-m+1) od: E:= exp(-x)/x^(m) * E: return(E); end: for n1 from 1 to nmax do f(n1-1) := simplify(exp(x) * x^(nmax+3) * EA(x, m, n1)); for m1 from 0 to nmax+2 do b(n1-1, m1) := coeff(f(n1-1), x, nmax+2-m1) od: od: for n1 from 0 to nmax-1 do for m1 from 0 to n1-m+1 do a(n1-m+2, m1+1) := abs(b(m1, n1-m1)) od: od: seq(seq(a(n, m), m = 1..n),n = 1..nmax-1);
# End program 2
# Maple programs revised by Johannes W. Meijer, Sep 22 2012

a(n,m) = (n-m+1)*a(n-1,m) + a(n-1,m-1), for 2 <= m <= n-1, with a(n,n) = 1 and a(n,1) = n*a(n-1,1) + (n-1)!.

a(n,m) = product(i, i= m..n)*sum(1/i, i = m..n).

A165675 Triangle read by rows. T(n, k) = (n - k + 1)! * H(k, n - k), where H are the hyperharmonic numbers. For 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 11, 5, 1, 24, 50, 26, 7, 1, 120, 274, 154, 47, 9, 1, 720, 1764, 1044, 342, 74, 11, 1, 5040, 13068, 8028, 2754, 638, 107, 13, 1, 40320, 109584, 69264, 24552, 5944, 1066, 146, 15, 1, 362880, 1026576, 663696, 241128, 60216, 11274, 1650, 191, 17, 1

Offset: 0

Johannes W. Meijer, Oct 05 2009

Previous name: Extended triangle related to the asymptotic expansions of the E(x, m = 2, n).
For the definition of the hyperharmonic numbers see the formula section.
This triangle is the same as triangle A165674 except for the extra left-hand column T(n, 0) = n!. The T(n) formulas for the right-hand columns generate the coefficients of this extra left-hand column.
Leroy Quet discovered triangle A105954 which is the reversal of our triangle.
In square format, row k gives the (n-1)-st elementary symmetric function of {k, k+1, k+2,..., k+n}, as in the Mathematica section. - Clark Kimberling, Dec 29 2011

			Triangle T(n, k) begins:
  [0]    1;
  [1]    1,     1;
  [2]    2,     3,    1;
  [3]    6,    11,    5,    1;
  [4]   24,    50,   26,    7,   1;
  [5]  120,   274,  154,   47,   9,   1;
  [6]  720,  1764, 1044,  342,  74,  11,  1;
  [7] 5040, 13068, 8028, 2754, 638, 107, 13, 1;
Seen as an array (the triangle arises when read by descending antidiagonals):
  [0] 1,  1,   2,    6,    24,    120,     720,     5040, ...
  [1] 1,  3,  11,   50,   274,   1764,   13068,   109584, ...
  [2] 1,  5,  26,  154,  1044,   8028,   69264,   663696, ...
  [3] 1,  7,  47,  342,  2754,  24552,  241128,  2592720, ...
  [4] 1,  9,  74,  638,  5944,  60216,  662640,  7893840, ...
  [5] 1, 11, 107, 1066, 11274, 127860, 1557660, 20355120, ...
  [6] 1, 13, 146, 1650, 19524, 245004, 3272688, 46536624, ...
  [7] 1, 15, 191, 2414, 31594, 434568, 6314664, 97053936, ...

A105954 is the reversal of this triangle.
A165674, A138771 and A165680 are related triangles.
A080663 equals the third right hand column.
A000142 equals the first left hand column.
A093345 are the row sums.
Columns include A165676, A165677, A165678 and A165679.

nmax := 8; for n from 0 to nmax do a(n, 0) := n! od: for n from 0 to nmax do a(n, n) := 1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (n-m+1)*a(n-1, m) + a(n-1, m-1) od: od: seq(seq(a(n, m), m=0..n), n=0..nmax);
# Johannes W. Meijer, revised Nov 27 2012
# Shows the array format, using hyperharmonic numbers.
H := proc(n, k) option remember; if n = 0 then 1/(k+1)
else add(H(n - 1, j), j = 0..k) fi end:
seq(lprint(seq((k + 1)!*H(n, k), k = 0..7)), n = 0..7);
# Shows the array format, using the hypergeometric formula.
A := (n, k) -> (k+1)*((n + k)! / n!)*hypergeom([-k, 1, 1], [2, n + 1], 1):
seq(lprint(seq(simplify(A(n, k)), k = 0..7)), n = 0..7);
# Peter Luschny, Jul 03 2022

a[n_] := SymmetricPolynomial[n - 1, t[n]]; z = 10;
t[n_] := Table[k - 1, {k, 1, n}]; t1 = Table[a[n], {n, 1, z}]  (* A000142 *)
t[n_] := Table[k,     {k, 1, n}]; t2 = Table[a[n], {n, 1, z}]  (* A000254 *)
t[n_] := Table[k + 1, {k, 1, n}]; t3 = Table[a[n], {n, 1, z}]  (* A001705 *)
t[n_] := Table[k + 2, {k, 1, n}]; t4 = Table[a[n], {n, 1, z}]  (* A001711 *)
t[n_] := Table[k + 3, {k, 1, n}]; t5 = Table[a[n], {n, 1, z}]  (* A001716 *)
t[n_] := Table[k + 4, {k, 1, n}]; t6 = Table[a[n], {n, 1, z}]  (* A001721 *)
t[n_] := Table[k + 5, {k, 1, n}]; t7 = Table[a[n], {n, 1, z}]  (* A051524 *)
t[n_] := Table[k + 6, {k, 1, n}]; t8 = Table[a[n], {n, 1, z}]  (* A051545 *)
t[n_] := Table[k + 7, {k, 1, n}]; t9 = Table[a[n], {n, 1, z}]  (* A051560 *)
t[n_] := Table[k + 8, {k, 1, n}]; t10 = Table[a[n], {n, 1, z}] (* A051562 *)
t[n_] := Table[k + 9, {k, 1, n}]; t11 = Table[a[n], {n, 1, z}] (* A051564 *)
t[n_] := Table[k + 10, {k, 1, n}];t12 = Table[a[n], {n, 1, z}] (* A203147 *)
t = {t1, t2, t3, t4, t5, t6, t7, t8, t9, t10};
TableForm[t]  (* A165675 in square format *)
m[i_, j_] := t[[i]][[j]];
(* A165675 as a sequence *)
Flatten[Table[m[i, n + 1 - i], {n, 1, 10}, {i, 1, n}]]
(* Clark Kimberling, Dec 29 2011 *)
A[n_, k_] := (k + 1)*((n + k)! / n!)*HypergeometricPFQ[{-k, 1, 1}, {2, n + 1}, 1];
Table[A[n, k], {n, 0, 7}, {k, 0, 7}] // TableForm (* Peter Luschny, Jul 03 2022 *)

from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0:
        return [1]
    row = Trow(n - 1) + [1]
    for m in range(n - 1, 0, -1):
        row[m] = (n - m + 1) * row[m] + row[m - 1]
    row[0] *= n
    return row
for n in range(9): print(Trow(n))  # Peter Luschny, Feb 27 2025

The hyperharmonic numbers are H(n, k) = Sum_{j=0..k} H(n - 1, j), with base condition H(0, k) = 1/(k + 1).
T(n, k) = (n - k + 1)*T(n - 1, k) + T(n - 1, k - 1), 1 <= k <= n-1, with T(n, 0) = n! and T(n, n) = 1.
From Peter Luschny, Jul 03 2022: (Start)
The rectangular array is given by:
A(n, k) = (k + 1)!*H(n, k).
A(n, k) = (k + 1)*((n + k)! / n!)*hypergeom([-k, 1, 1], [2, n + 1], 1). (End)
From Werner Schulte, Feb 26 2025: (Start)
T(n, k) = n * T(n-1, k) + (n-1)! / (k-1)! for 0 < k < n.
T(n, k) = (Sum_{i=k..n} 1/i) * n! / (k-1)! for 0 < k <= n.
Matrix inverse M = T^(-1) is given by: M(n, n) = 1, M(n, n-1) = 1 - 2 * n for n > 0, M(n, n-2) = (n-1)^2 for n > 1, and M(i, j) = 0 otherwise. (End)

New name from Peter Luschny, Jul 03 2022

A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!x^i(1+x)^(n-i)/(n+1-i).

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 2, 7, 11, 0, 6, 26, 46, 50, 0, 24, 126, 274, 326, 274, 0, 120, 744, 1956, 2844, 2556, 1764, 0, 720, 5160, 16008, 28092, 30708, 22212, 13068, 0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584, 0, 40320

Offset: 1

N. J. A. Sloane and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

The first nonzero column gives the factorial numbers, which are Stirling_1(*,1), the rightmost diagonal gives Stirling_1(*,2), so this triangle may be regarded as interpolating between the first two columns of the Stirling numbers of the first kind.
This is a slice (the right-hand wall) through the infinite square pyramid described in the link. The other three walls give A007318 and A008276 (twice).
The coefficients of the A165674 triangle are generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). The a(n) formulas for the coefficients in the right hand columns of this triangle lead to Wiggen's triangle A028421 and their o.g.f.s. lead to the sequence given above. Some right hand columns of the A165674 triangle are A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009

			Triangle begins:
0,
0, 1,
0, 1, 3,
0, 2, 7, 11,
0, 6, 26, 46, 50,
0, 24, 126, 274, 326, 274,
0, 120, 744, 1956, 2844, 2556, 1764,
0, 720, 5160, 16008, 28092, 30708, 22212, 13068,
0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584,
0, 40320, 367920, 1498320, 3582000, 5562576, 5868144, 4292496, 2239344, 1026576, ...

N. J. A. Sloane, Notes on Carlo Wood's Polynomials

Columns give A000142, A108217, A126672; diagonals give A000254, A067318, A126673. Row sums give A126674. Alternating row sums give A000142.
See A126682 for the full pyramid of coefficients of the underlying polynomials.
Cf. A165674, A028421, A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009

for n from 1 to 15 do t1:=add( n!*x^i*(1+x)^(n-i)/(n+1-i), i=1..n); series(t1,x,100); lprint(seriestolist(%)); od:

Join[{{0}}, Reap[For[n = 1, n <= 15, n++, t1 = Sum[n!*x^i*(1+x)^(n-i)/(n+1-i), {i, 1, n}]; se = Series[t1, {x, 0, 100}]; Sow[CoefficientList[se, x]]]][[2, 1]]] // Flatten (* Jean-François Alcover, Jan 07 2014, after Maple *)

Recurrence: T(n,0) = 0; for n>=0, i>=1, T(n+1,i) = (n+1)*T(n,i) + n!*binomial(n,i).

E.g.f.: x*log(1-(1+x)*y)/(x*y-1)/(1+x). - Vladeta Jovovic, Feb 13 2007

A105954 Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 11, 6, 1, 7, 26, 50, 24, 1, 9, 47, 154, 274, 120, 1, 11, 74, 342, 1044, 1764, 720, 1, 13, 107, 638, 2754, 8028, 13068, 5040, 1, 15, 146, 1066, 5944, 24552, 69264, 109584, 40320, 1, 17, 191, 1650, 11274, 60216, 241128, 663696, 1026576, 362880

Offset: 0

Leroy Quet, Jun 26 2005

Antidiagonal sums are A093345 (n! * (1 + Sum_{i=1..n}((1/i)*Sum_{j=0..i-1} 1/j!))). - Gerald McGarvey, Aug 27 2005
A recasting of A093905 and A067176. - R. J. Mathar, Mar 01 2009
The triangular array of this sequence is the reversal of A165675 which is related to the asymptotic expansion of the higher order exponential integral E(x,m=2,n); see also A165674. - Johannes W. Meijer, Oct 16 2009

			A(2, 2) = (1 + (1 + 1/2) + (1 + 1/2 + 1/3))*6 = 26.
Array A(n, k) begins:
  [n\k]  0       1       2        3        4        5          6
  -------------------------------------------------------------------
  [0]    1,      1,      1,       1,       1,       1,         1, ...
  [1]    1,      3,      5,       7,       9,       11,       13, ...
  [2]    2,     11,     26,      47,      74,      107,      146, ...
  [3]    6,     50,    154,     342,     638,     1066,     1650, ...
  [4]   24,    274,   1044,    2754,    5944,    11274,    19524, ...
  [5]  120,   1764,   8028,   24552,   60216,   127860,   245004, ...
  [6]  720,  13068,  69264,  241128,  662640,  1557660,  3272688, ...
  [7] 5040, 109584, 663696, 2592720, 7893840, 20355120, 46536624, ...

G. C. Greubel, Table of n, a(n) for the first 27 rows, flattened
Arthur T. Benjamin, David Gaebler and Robert Gaebler, A Combinatorial Approach to Hyperharmonic Numbers, INTEGERS, Electronic Journal of Combinatorial Number Theory, Volum 3, #A15, 2003.

Column 0 = A000142 (factorial numbers).
Column 1 = A000254 (Stirling numbers of first kind s(n, 2)) starting at n=1.
Column 2 = A001705 (Generalized Stirling numbers: a(n) = n!*Sum_{k=0..n-1}(k+1)/(n-k)), starting at n=1.
Column 3 = A001711 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*3^k*stirling1(n+1, k+1)).
Column 4 = A001716 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1)).
Column 5 = A001721 (Generalized Stirling numbers: a(n) = Sum_{k=0..n}(-1)^(n+k)*binomial(k+1, 1)*5^k*stirling1(n+1, k+1)).
Column 6 = A051524 (2nd unsigned column of A051338) starting at n=1.
Column 7 = A051545 (2nd unsigned column of A051339) starting at n=1.
Column 8 = A051560 (2nd unsigned column of A051379) starting at n=1.
Column 9 = A051562 (2nd unsigned column of A051380) starting at n=1.
Column 10= A051564 (2nd unsigned column of A051523) starting at n=1.
2nd row is A005408 (2n - 1, starting at n=1).
3rd row is A080663 (3n^2 - 1, starting at n=1).
Main diagonal gives A384024.
Cf. A000254, A165674 and A165675, A028421 and A126671.

H := proc(n, k) option remember; if n = 0 then 1/k else add(H(n - 1, j), j = 1..k) fi end: A := (n, k) -> (n + 1)!*H(k, n + 1):
# Alternative with standard harmonic number:
A := (n, k) -> if k = 0 then n! else (harmonic(n + k) - harmonic(k - 1))*(n + k)! / (k - 1)! fi:
for n from 0 to 7 do seq(A(n, k), k = 0..6) od;
# Alternative with hypergeometric formula:
A := (n, k) -> (n+1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k+1], 1):
seq(print(seq(simplify(A(n, k)), k = 0..6)), n=0..7); # Peter Luschny, Jul 01 2022

H[0, m_] := 1/m; H[n_, m_] := Sum[H[n - 1, k], {k, m}]; a[n_, m_] := m!H[n, m]; Flatten[ Table[ a[i, n - i], {n, 10}, {i, n - 1, 0, -1}]]
Table[ a[n, m], {m, 8}, {n, 0, m + 1}] // TableForm (* to view the table *)
(* Robert G. Wilson v, Jun 27 2005 *)

a(n, k) = polcoef(prod(j=0, n, 1+(j+k)*x), n); \\ Seiichi Manyama, May 19 2025

A(n, k) = (Harmonic(n + k) - Harmonic(k - 1))*(n + k)!/(k - 1)! if k > 0, otherwise n!.
From Gerald McGarvey, Aug 27 2005, edited by Peter Luschny, Jul 02 2022: (Start)
E.g.f. for column k: -log(1 - x)/(x*(1 - x)^k).
Row 3 is r(n) = 4*n^3 + 18*n^2 + 22*n + 6.
Row 4 is r(n) = 5*n^4 + 40*n^3 + 105*n^2 + 100*n + 24.
Row 5 is r(n) = 6*n^5 + 75*n^4 + 340*n^3 + 675*n^2 + 548*n + 120.
Row 6 is r(n) = 7*n^6 + 126*n^5 + 875*n^4 + 2940*n^3 + 4872*n^2 + 3528*n + 720.
Row 7 is r(n) = 8*n^7 + 196*n^6 + 1932*n^5 + 9800*n^4 + 27076*n^3 + 39396*n^2 + 26136*n + 5040.
The sum of the polynomial coefficients for the n-th row is |S1(n, 2)|, which are the unsigned Stirling1 numbers which appear in column 1.
A(m, n) = Sum_{k=1..m} n*A094645(m, n)*(n+1)^(k-1). (A094645 is Generalized Stirling number triangle of first kind, e.g.f.: (1-y)^(1-x).) (End)
In Gerard McGarvey's formulas for the row coefficients we find Wiggen's triangle A028421 and their o.g.f.s lead to Wood's polynomials A126671; see A165674. - Johannes W. Meijer, Oct 16 2009
A(n, k) = (n + 1)*((n + k)! / k!)*hypergeom([-n, 1, 1], [2, k + 1], 1). - Peter Luschny, Jul 01 2022
A(n,k) = [x^n] Product_{j=0..n} (1 + (j+k)*x). - Seiichi Manyama, May 19 2025

More terms from Robert G. Wilson v, Jun 27 2005

Edited by Peter Luschny, Jul 02 2022

A121670 a(n) = n^3 - 3*n.

Original entry on oeis.org

0, -2, 2, 18, 52, 110, 198, 322, 488, 702, 970, 1298, 1692, 2158, 2702, 3330, 4048, 4862, 5778, 6802, 7940, 9198, 10582, 12098, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 32672, 35838, 39202, 42770, 46548, 50542, 54758, 59202, 63880, 68798, 73962

Offset: 0

Gary W. Adamson, Aug 14 2006

Previous name was: Real part of (n + i)^3, companion to A080663.
Reversing the order of terms in (n + i)^3 to (1 + ni)^3 generates the terms of A080663. E.g, A080663(4) = 47 since (1 + 4i)^3 = (-47 - 52i). Or, (n + i)^3 = (a(n) + A080663(a)i) and (1 + ni)^3 = (-A080663(n) - a(n)i).
Also, numbers n such that the polynomial x^6 - n*x^3 + 1 is reducible. - Ralf Stephan, Oct 24 2013

			a(4) = 52 since (4 + i)^3 = (52 + 47i); where 47 = A080663(4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A058794, A080663.

CoefficientList[Series[-2 x (x^2 - 5 x + 1)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 11 2014 *)
Table[n^3-3n,{n,0,60}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,-2,2,18},60] (* Harvey P. Dale, Nov 30 2021 *)

Vec(-2*x*(x^2-5*x+1)/(x-1)^4 + O(x^100)); \\ Colin Barker, Oct 16 2013

a(n) = Re( (n + i)^3 ).
a(n) = n^3-3*n. a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: -2*x*(x^2-5*x+1) / (x-1)^4. - Colin Barker, Oct 16 2013
a(n)^2 = A028872(n)^3 + 3*A028872(n)^2 for n>1. - Bruno Berselli, May 03 2018
a(n) = A058794(n-2) for n>1. - Altug Alkan, May 03 2018

Terms corrected, new name, and more terms from Colin Barker, Oct 16 2013

A089682 Primes of the form 3*m^2 - 1.

Original entry on oeis.org

2, 11, 47, 107, 191, 431, 587, 971, 1451, 2027, 2351, 2699, 3467, 4799, 5807, 6911, 7499, 8111, 8747, 10091, 10799, 14699, 15551, 16427, 17327, 18251, 25391, 27647, 36299, 41771, 44651, 55487, 57131, 62207, 67499, 71147, 74891, 80687, 92927, 99371, 103787, 106031

Offset: 1

Giovanni Teofilatto, Jan 05 2004

431 and 27647 also have the form 2*m^3-1 (431 = 3*12^2-1 = 2*6^3-1 and 27647 = 3*96^2-1 = 2*24^3-1). [Howard Berman (howard_berman(AT)hotmail.com), May 09 2009]
Or, primes p such that 3*(p+1) is a square. [Vincenzo Librandi, Nov 18 2010]
Obviously m must be either 1 or an even number. So this consists of 1 together with primes of the form 12*u^2 - 1. - N. J. A. Sloane, Aug 09 2019

M. Cerasoli, F. Eugeni, and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Primes in A080663.

Cf. A089681.

[a: n in [0..200] | IsPrime(a) where a is 3*n^2-1]; // Vincenzo Librandi, Jul 16 2012

Select[Table[3 n^2 - 1, {n, 0, 2000}], PrimeQ] (* Vincenzo Librandi, Jul 16 2012 *)

a(n) = A080663(A089681(n)). - Elmo R. Oliveira, Feb 22 2025

More terms from Rick L. Shepherd, May 18 2005

A158463 a(n) = 12*n^2 - 1.

Original entry on oeis.org

-1, 11, 47, 107, 191, 299, 431, 587, 767, 971, 1199, 1451, 1727, 2027, 2351, 2699, 3071, 3467, 3887, 4331, 4799, 5291, 5807, 6347, 6911, 7499, 8111, 8747, 9407, 10091, 10799, 11531, 12287, 13067, 13871, 14699, 15551, 16427, 17327, 18251, 19199, 20171, 21167, 22187

Offset: 0

Vincenzo Librandi, Mar 19 2009

Related to Legendre polynomials - see Mma line. - N. J. A. Sloane, Nov 17 2009
One notices that this sequence produces an inordinate number of semiprimes, perhaps better than mere chance for large values of n. - J. M. Bergot, Jun 30 2011
Sequence found by reading the line from -1, in the direction -1, 11, ..., in the square spiral whose vertices are -1 together with the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

			G.f. = -1 + 11*x + 47*x^2 + 107*x^3 + 191*x^4 + 299*x^5 + 431*x^6 + 587*x^7 + 767*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001082, A080663, A135453.

[12*n^2 - 1: n in [0..100]]; // G. C. Greubel, Sep 25 2018

Table[Numerator[LegendreP[2,2n]],{n,0,50}] (* N. J. A. Sloane, Nov 17 2009 *)
a[ n_] := 2 LegendreP[ 2, 2 n] (* Michael Somos, Jul 01 2011 *)
12*Range[0,40]^2-1 (* or *) LinearRecurrence[{3,-3,1},{-1,11,47},50] (* Harvey P. Dale, Jun 22 2019 *)

{a(n) = 12 * n^2 - 1} /* Michael Somos, Nov 12 2011 */

a(-n) = a(n).
G.f.: (1-14*x-11*x^2)/(x-1)^3. - R. J. Mathar, Aug 27 2011
a(n) = A135453(n) - 1. - Omar E. Pol, Jul 18 2012
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(12))*cot(Pi/sqrt(12)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(12))*csc(Pi/sqrt(12)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(12))*csc(Pi/sqrt(12)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(12))*sin(Pi/sqrt(6))/sqrt(2). (End)
From Gerry Martens, Apr 06 2024: (Start)
a(n) = Re((2*n*i-1)^3).
a(n) = -8*(1/4+n^2)^(3/2)*cos(3*arctan(2*n)). (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(12*x^2 + 12*x - 1).
a(n) = A080663(2*n) for n > 0. (End)

Minor edits by N. J. A. Sloane, Nov 17 2009

Comment rewritten by Bruno Berselli, Aug 27 2011

A188386 a(n) = numerator(H(n+2)-H(n-1)), where H(n) = Sum_{k=1..n} 1/k is the n-th harmonic number.

Original entry on oeis.org

11, 13, 47, 37, 107, 73, 191, 121, 299, 181, 431, 253, 587, 337, 767, 433, 971, 541, 1199, 661, 1451, 793, 1727, 937, 2027, 1093, 2351, 1261, 2699, 1441, 3071, 1633, 3467, 1837, 3887, 2053, 4331, 2281, 4799, 2521, 5291, 2773, 5807, 3037, 6347, 3313, 6911, 3601

Offset: 1

Gary Detlefs, Mar 29 2011

Denominators are listed in A033931.
A027446 appears to be divisible by a(n).
The sequence lists also the largest odd divisors of 3*m^2-1 (A080663) for m>1. In fact, for m even, the largest odd divisor is 3*m^2-1 itself; for m odd, the largest odd divisor is (3*m^2-1)/2. From this follows the second formula given in Formula field. - Bruno Berselli, Aug 27 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A033931 (denominators), A001008, A001711, A027446, A002805, A003154, A080663, A158463, A213998.

import Data.Ratio ((%), numerator)
a188386 n = a188386_list !! (n-1)
a188386_list = map numerator $ zipWith (-) (drop 3 hs) hs
   where hs = 0 : scanl1 (+) (map (1 %) [1..])
-- Reinhard Zumkeller, Jul 03 2012

[Numerator((3*n^2+6*n+2)/((n*(n+1)*(n+2)))): n in [1..50]]; // Vincenzo Librandi, Mar 30 2011

seq((3-(-1)^n)*(3*n^2+6*n+2)/4, n=1..100);

Table[(3 - (-1)^n)*(3*n^2 + 6*n + 2)/4, {n, 40}] (* Wesley Ivan Hurt, Jan 29 2017 *)
Numerator[#[[4]]-#[[1]]]&/@Partition[HarmonicNumber[Range[0,50]],4,1] (* or *) LinearRecurrence[{0,3,0,-3,0,1},{11,13,47,37,107,73},50] (* Harvey P. Dale, Dec 31 2017 *)

a(n) = numerator((3*n^2+6*n+2)/(n*(n+1)*(n+2))).
a(n) = (3-(-1)^n)*(3*n^2+6*n+2)/4.
a(2n+1) = A158463(n+1), a(2n) = A003154(n+1).
G.f.: -x*(11+13*x+14*x^2-2*x^3-x^4+x^5) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Apr 09 2011
a(n) = numerator of coefficient of x^3 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - Francesco Daddi, Aug 04 2011
H(n+3) = 3/2 + 2*f(n)/((n+2)*(n+3)), where f(n) = Sum_{k=0..n}((-1)^k*binomial(-3,k)/(n+1-k)). - Gary Detlefs, Jul 17 2011
a(n) = A213998(n+2,2). - Reinhard Zumkeller, Jul 03 2012
Sum_{n>=1} 1/a(n) = c*(tan(c) - cot(c)/2) - 1/2, where c = Pi/(2*sqrt(3)). - Amiram Eldar, Sep 27 2022

A271713 Numbers n such that 3*n - 5 is a square.

Original entry on oeis.org

2, 3, 7, 10, 18, 23, 35, 42, 58, 67, 87, 98, 122, 135, 163, 178, 210, 227, 263, 282, 322, 343, 387, 410, 458, 483, 535, 562, 618, 647, 707, 738, 802, 835, 903, 938, 1010, 1047, 1123, 1162, 1242, 1283, 1367, 1410, 1498, 1543, 1635, 1682, 1778, 1827, 1927, 1978, 2082, 2135, 2243, 2298

Offset: 1

Juri-Stepan Gerasimov, Apr 12 2016

Quasipolynomial of order 2 and degree 2. - Charles R Greathouse IV, Apr 12 2016
From Ray Chandler, Apr 13 2016: (Start)
Square roots of resulting squares gives A001651.
Sequence is the union of A141631 and A271740. (End)

			a(3) = 7 because 3*7 - 5 = 16 = 4^2.

Index entries for linear recurrences with constant coefficients, signature (1, 2, -2, -1, 1).

Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), this sequence (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), A271675 (k=4), A100536 (k=6).

Cf. A001651, A141631, A271740.

[ n: n in [0..2500] | IsSquare(3*n - 5)];

A271713:=n->`if`(issqr(3*n-5), n, NULL): seq(A271713(n), n=1..5000); # Wesley Ivan Hurt, Apr 13 2016

Select[Range@ 2300, IntegerQ@ Sqrt[3 # - 5] &] (* Michael De Vlieger, Apr 12 2016 *)

is(n)=issquare(3*n-5) \\ Charles R Greathouse IV, Apr 12 2016

a(n)=((n\2*3-(-1)^n)^2+5)/3 \\ Charles R Greathouse IV, Apr 12 2016

from _future_ import division
A271713_list = [(n**2+5)//3 for n in range(10**6) if not (n**2+5) % 3] # Chai Wah Wu, Apr 13 2016

G.f.: x*(2 + x + x^3 + 2*x^4)/((1 - x)^3*(1 + x)^2). - Ilya Gutkovskiy, Apr 12 2016
a(n) = (3/2)*n^2 + O(n). - Charles R Greathouse IV, Apr 12 2016
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5. - Wesley Ivan Hurt, Apr 13 2016

A121672 Real part of (n + i)^5.

Original entry on oeis.org

0, -4, -38, -12, 404, 1900, 5646, 13412, 27688, 51804, 90050, 147796, 231612, 349388, 510454, 725700, 1007696, 1370812, 1831338, 2407604, 3120100, 3991596, 5047262, 6314788, 7824504, 9609500, 11705746, 14152212, 16990988, 20267404, 24030150

Offset: 0

Gary W. Adamson, Aug 14 2006

The companion sequence A121671 uses the operation (1 + n*i)^5.

			a(4) = 404 since (4 + i)^5 = (404 + 1121i) where 1121 = A121671(4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A121671, A080663, A121670.

Re[(Range[0,30]+I)^5] (* Harvey P. Dale, Mar 01 2012 *)

a(n) = real((n + I)^5); \\ Michel Marcus, Dec 19 2020

G.f.: -2*x*(2+7*x-78*x^2+7*x^3+2*x^4)/(1-x)^6. - Bruno Berselli, Mar 01 2012
a(n) = n*(n^4-10*n^2+5). - Bruno Berselli, Mar 01 2012
a(n) = (1+n^2)^(5/2)*sin(5*arctan(n)). - Gerry Martens, Apr 06 2024

Corrected and extended by Harvey P. Dale, Mar 01 2012

cp's OEIS Frontend

A165674 Triangle generated by the asymptotic expansions of the E(x,m=2,n).

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A165675 Triangle read by rows. T(n, k) = (n - k + 1)! * H(k, n - k), where H are the hyperharmonic numbers. For 0 <= k <= n.

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A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!*x^i*(1+x)^(n-i)/(n+1-i).

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A105954 Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j), for 0 <= k <= n.

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A121670 a(n) = n^3 - 3*n.

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A089682 Primes of the form 3*m^2 - 1.

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Magma

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A158463 a(n) = 12*n^2 - 1.

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A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!x^i(1+x)^(n-i)/(n+1-i).