A003633 -id:A003633 - cp's OEIS frontend

A001662 Coefficients of Airey's converging factor.

0, 1, 1, -1, -1, 13, -47, -73, 2447, -16811, -15551, 1726511, -18994849, 10979677, 2983409137, -48421103257, 135002366063, 10125320047141, -232033147779359, 1305952009204319, 58740282660173759, -1862057132555380307, 16905219421196907793, 527257187244811805207

Offset: 0

N. J. A. Sloane

A051711 times the coefficient in expansion of W(exp(x)) about x=1, where W is the Lambert function. - Paolo Bonzini, Jun 22 2016

The polynomials with coefficients in triangle A008517, evaluated at -1.

			G.f. = x + x^2 - x^3 - x^4 + 13*x^5 - 47*x^6 - 73*x^7 + 2447*x^8 + ... - _Michael Somos_, Jun 23 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms 29 onwards updated by Sean A. Irvine, April 25 2019)
J. R. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions, Phil. Mag., 24 (1937), 521-552 [ gives 22 terms ].
J. A. Airey, The "converging factor" in asymptotic series and the calculation of Bessel, Laguerre and other functions [Annotated scanned copy]
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
J. M. Borwein and R. M. Corless, Emerging tools for experimental mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W Function, Advances in Computational Mathematics, (5), 1996, pp. 329-359.
R. M. Corless, D. J. Jeffrey and D. E. Knuth, A sequence of series for the Lambert W Function (section 2.2).
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
Vaclav Kotesovec, Graph - the asymptotic ratio
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
J. C. P. Miller, A method for the determination of converging factors ..., Proc. Camb. Phil. Soc., 48 (1952), 243-254.
J. C. P. Miller, A method for the determination of converging factors ... [Annotated scanned copy]
F. D. Murnaghan, Airey's converging factor, Proc. Nat. Acad. Sci. USA, 69 (1972), 440-441.
F. D. Murnaghan and J. W. Wrench, Jr., The Converging Factor for the Exponential Integral, Report 1535, David Taylor Model Basin, U.S. Dept. of Navy, 1963 [ gives first 67 terms ].
N. J. A. Sloane, Letter to F. D. Murnaghan, Apr 17, 1974
J. W. Wrench, Jr., Letter to N. J. A. Sloane, 24 Apr 1974
P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument I A, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 721-736.
P. Wynn, Converging factors for the Weber parabolic cylinder functions of complex argument I B, Proc. Konin. Ned. Akad. Weten., Series A, 66 (1963), 737-754.
P. Wynn, Converging factors for the Weber parabolic cylinder functions ... [Annotated scan of part 2 only]

Cf. A051711, A032188, A274447, A274448, A340556.

with(combinat); A001662 := proc(n) add((-1)^k*eulerian2(n-1,k),k=0..n-1) end:
seq(A001662(i),i=0..23); # Peter Luschny, Nov 13 2012

a[0] = 0; a[n_] := Sum[ (n+k-1)! * Sum[ (-1)^j/(k-j)! * Sum[ 1/i! * StirlingS1[n-i+j-1, j-i] / (n-i+j-1)!, {i, 0, j}] * 2^(n-j-1), {j, 0, k}], {k, 0, n-1}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 26 2013, after Vladimir Kruchinin *)
a[ n_] := If[ n < 1, 0, 2^(n - 1) Sum[ (-2)^-j StirlingS1[n - i + j - 1, j - i] Binomial[n + k - 1, n + j - 1] Binomial[n + j - 1, i], {k, 0, n - 1}, {j, 0, k}, {i, 0, j}]]; (* Michael Somos, Jun 23 2019 *)
len := 12; gf := (1/2) (LambertW[Exp[x + 1]] - 1);
ser := Series[gf, { x, 0, len}]; norm := Table[n! 4^n, {n, 0, len}];
CoefficientList[ser, x] * norm (* Peter Luschny, Jun 24 2019 *)

a(n):= if n=0 then 1 else (sum((n+k-1)!*sum(((-1)^(j)/(k-j)!*sum((1/i! *stirling1(n-i+j-1, j-i))/(n-i+j-1)!, i, 0, j))*2^(n-j-1), j, 0, k), k, 0, n-1)); /* Vladimir Kruchinin, Nov 11 2012 */

@CachedFunction
def eulerian2(n, k):
    if k==0: return 1
    elif k==n: return 0
    return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
def A001662(n): return add((-1)^k*eulerian2(n-1,k) for k in (0..n-1))
[A001662(m) for m in (0..23)] # Peter Luschny, Nov 13 2012

Let b(n) = 0, 1, -1, 1, 1, -13,.. be the sequence with all signs but one reversed: b(1)=a(1), b(n)=-a(n) for n<>1. Define the e.g.f. B(x) = 2*Sum_{n>=0} b(n)*(x/2)^n/n!. B(x) satisfies exp(B(x)) = 1 + 2*x - B(x). [Bernstein/Sloane S52]
Similarly, c(0)=1, c(n)=-a(n+1) are the alternating row sums of the second-order Eulerian numbers A340556, or c(n) = E2poly(n,-1). - Peter Luschny, Feb 13 2021
a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^j/(k-j)!)*Sum_{i=0..j} ((1/i!)*Stirling1(n-i+j-1,j-i)/(n-i+j-1)!)*2^(n-j-1), n > 0, a(0)=1. - Vladimir Kruchinin, Nov 11 2012
From Sergei N. Gladkovskii, Nov 24 2012, Aug 22 2013: (Start)
Continued fractions:
G.f.: 2*x - x/G(0) where G(k) = 1 - 2*x*k + x*(k+1)/G(k+1).
G.f.: 2*x - 2*x/U(0) where U(k) = 1 + 1/(1 - 4*x*(k+1)/U(k+1)).
G.f.: A(x) = x/G(0) where G(k) = 1 - 2*x*(k+1) + x*(k+1)/G(k+1).
G.f.: 2*x  - x*W(0) where W(k) = 1 + x*(2*k+1)/( x*(2*k+1) + 1/(1 + x*(2*k+2)/( x*(2*k+2) + 1/W(k+1)))). (End)
a(n) = 4^n * Sum_{i=1..n} Stirling2(n,i)*A013703(i)/2^(2*i+1). - Paolo Bonzini, Jun 23 2016
E.g.f.: 1/2*(LambertW(exp(4*x+1))-1). - Vladimir Kruchinin, Feb 18 2018
a(0) = 0; a(1) = 1; a(n) = 2 * a(n-1) - Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020

More terms from James Sellers, Dec 07 1999

Reverted to converging factors definition by Paolo Bonzini, Jun 23 2016

A010739 Shifts 2 places left under inverse binomial transform.

Original entry on oeis.org

1, 2, 1, 1, -2, 3, -7, 22, -71, 231, -794, 2945, -11679, 48770, -212823, 969221, -4605674, 22802431, -117322423, 625743878, -3452893503, 19684083947, -115787084242, 701935339725, -4380330298815, 28105726916034, -185229395693615, 1252696143653513

Offset: 0

N. J. A. Sloane, Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..650
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A010741, A010743, A010745, A010747.

a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-2)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022

a[n_] := a[n] = Function[m, If[m<0, 2^n,
   Sum[a[m-j]*Binomial[m, j]*(-1)^j, {j, 0, m}]]][n-2];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jul 24 2022, after Alois P. Heinz *)

G.f. A(x) satisfies: A(x) = 1 + 2*x + x^2*A(x/(1 + x))/(1 + x). - Ilya Gutkovskiy, Feb 02 2022

A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 9, 3, 2, 1, 1, 20, 6, 3, 2, 1, 1, 48, 10, 5, 3, 2, 1, 1, 115, 20, 8, 5, 3, 2, 1, 1, 286, 36, 14, 7, 5, 3, 2, 1, 1, 719, 72, 23, 12, 7, 5, 3, 2, 1, 1, 1842, 137, 40, 18, 11, 7, 5, 3, 2, 1, 1, 4766, 275, 69, 30, 16, 11, 7, 5, 3, 2, 1, 1, 12486, 541, 121, 47, 25, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Alois P. Heinz, Sep 07 2008

			T(5,1) = ([1,2,4]*[1,1,4] + [1]*[1]*4 + [1,2]*[1,1]*2 + [1,3]*[1,2]*1)/4 = 36/4 = 9.
Triangle begins:
    1;
    1,  1;
    2,  1,  1;
    4,  2,  1,  1;
    9,  3,  2,  1, 1;
   20,  6,  3,  2, 1, 1;
   48, 10,  5,  3, 2, 1, 1;
  115, 20,  8,  5, 3, 2, 1, 1;
  286, 36, 14,  7, 5, 3, 2, 1, 1;
  719, 72, 23, 12, 7, 5, 3, 2, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027.
T(2n,n) gives A000041(n).
Cf. A316074.

etrk:= proc(p) proc(n, k) option remember; `if`(n=0, 1,
         add(add(d*p(d, k), d=numtheory[divisors](j))*
         procname(n-j, k), j=1..n)/n)
       end end:
B:= etrk(T):
T:= (n, k)-> `if`(n<=k, `if`(n=0, 0, 1), B(n-k, k)):
seq(seq(T(n, k), k=1..n), n=1..14);

etrk[p_] := Module[{f}, f[n_, k_] := f[n, k] = If[n == 0, 1, (Sum[Sum[d*p[d, k], {d, Divisors[j]}]*f[n-j, k], {j, 1, n-1}] + Sum[d*p[d, k], {d, Divisors[n]}])/n]; f]; b = etrk[t]; t[n_, k_] := If[n <= k, If[n == 0, 0, 1], b[n-k, k]]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)

A279215 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k+1)/6).

Original entry on oeis.org

1, 1, 6, 20, 65, 190, 571, 1616, 4555, 12439, 33515, 88517, 230738, 592321, 1502384, 3763946, 9328899, 22880511, 55585077, 133806273, 319373068, 756124040, 1776497540, 4143489680, 9597505006, 22083821765, 50494638926, 114758996621, 259303832735, 582655202940, 1302234303910, 2895530963661, 6406348746390

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the square pyramidal numbers (A000330).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A000335, A279216, A279217, A279218, A279219.

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).

a(n) ~ exp(Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(24883200000*Zeta(5)^3) + Pi^8*Zeta(3)/(1728000*Zeta(5)^2) - Zeta(3)^2/(720*Zeta(5)) + Zeta'(-3)/3 + (Pi^12/(43200000*2^(3/5)*Zeta(5)^(11/5)) - Pi^4*Zeta(3) / (3600*2^(3/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(144000*2^(1/5)*Zeta(5)^(7/5)) + Zeta(3)/(12*2^(1/5)*Zeta(5)^(2/5))) * n^(2/5) + Pi^4/(180*2^(4/5)*Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5)/2^(7/5) * n^(4/5)) * Zeta(5)^(23/225) / (2^(29/150) * sqrt(5*Pi) * n^(271/450)). - Vaclav Kotesovec, Dec 08 2016

A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

Original entry on oeis.org

1, 1, 7, 25, 86, 269, 862, 2606, 7812, 22704, 64989, 182356, 504414, 1373694, 3693367, 9804435, 25733084, 66808578, 171719539, 437183839, 1103143657, 2760037810, 6850400668, 16873338215, 41260373472, 100196920196, 241712863504, 579416535973, 1380517695672, 3270075208145, 7702580246941

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the pentagonal pyramidal numbers (A002411).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000335, A002411, A279215, A279217, A279218, A279219.

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

a(n) ~ exp(-Zeta(3)/(8*Pi^2) - Pi^16/(83980800000*Zeta(5)^3) + Zeta'(-3)/2 + (Pi^12/(97200000*2^(2/5)*3^(1/5)*Zeta(5)^(11/5))) * n^(1/5) + (-Pi^8/(108000*2^(4/5)*3^(2/5)*Zeta(5)^(7/5))) * n^(2/5) + (Pi^4/(180*2^(1/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(8/5))) * n^(4/5)) * (3*Zeta(5))^(119/1200) / (2^(181/600) * sqrt(5*Pi) * n^(719/1200)). - Vaclav Kotesovec, Dec 08 2016

A279361 Exponential transform of the triangular numbers.

Original entry on oeis.org

1, 1, 4, 16, 80, 471, 3127, 23059, 186468, 1635265, 15422471, 155388399, 1663294756, 18826525771, 224434810797, 2808247979611, 36770685485408, 502505495269521, 7150461569849395, 105723461155720879, 1621191824611307436, 25738508587975433251

Offset: 0

Ilya Gutkovskiy, Dec 10 2016

nonn

			E.g.f.: A(x) = 1 + x/1! + 4*x^2/2! + 16*x^3/3! + 80*x^4/4! + 471*x^5/5! + 3127*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..519
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Exponential Transform
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A033462.

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*j*(j+1)/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 11 2016

Range[0, 23]! CoefficientList[Series[Exp[Exp[x] x (x + 2)/2], {x, 0, 23}], x]

E.g.f.: exp(exp(x)*x*(x+2)/2).

A007460 Shifts left under OR-convolution with itself.

Original entry on oeis.org

1, 1, 2, 7, 20, 58, 174, 519, 1550, 4634, 13884, 41616, 124824, 374390, 1123288, 3369297, 10107324, 30320434, 90961626, 272878138, 818632094, 2455888346, 7367661682, 22102935920, 66308767426, 198926187730, 596778527246, 1790335274112, 5371006016314

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..2097
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

import Data.Bits ((.|.))
a007460 n = a007460_list !! n
a007460_list = 1 : f [1,1] where
   f xs = x : f (x:xs) where
     x = sum $ zipWith (.|.) xs $ tail $ reverse xs :: Integer
-- Reinhard Zumkeller, Dec 29 2012

a:= proc(n) option remember; `if`(n=0, 1, add(
      Bits[Or](a(i), a(n-1-i)), i=0..n-1))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 22 2012, revised, Jun 16 2018

a[0]=1; a[1]=1; a[n_] := a[n] = Sum[BitOr[a[k], a[n-k-1]], {k, 0, n-1}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 07 2012, after Alois P. Heinz *)

a(n) ~ c * 3^n, where c = 0.23477965293553321256228091184896942343087043274059369777596946751928557... . - Vaclav Kotesovec, Sep 11 2014

A007464 Shifts left under GCD-convolution with itself.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 11, 10, 18, 16, 20, 24, 26, 20, 45, 40, 38, 34, 62, 46, 54, 50, 84, 50, 102, 78, 104, 98, 90, 70, 189, 82, 130, 84, 120, 112, 130, 120, 232, 152, 234, 132, 130, 208, 282, 140, 462, 180, 210, 220, 418, 284, 334, 260, 520, 156, 334, 556

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A178063 (partial sums).

a007464 n = a007464_list !! n
a007464_list = 1 : 1 : f [1,1] where
   f xs = y : f (y:xs) where y = sum $ zipWith gcd xs $ reverse xs
-- Reinhard Zumkeller, Jan 21 2014

a:= proc(n) option remember;
      `if`(n=0, 1, add(igcd(a(i), a(n-1-i)), i=0..n-1))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 22 2012

a[0]=1; a[1]=1; a[n_] := a[n] = Sum[GCD[a[k], a[n-k-1]], {k, 0, n-1}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 07 2012, after Alois P. Heinz *)

N=66; v=vector(N);  v[1]=1; for(n=2,N, v[n]=sum(k=1,n-1, gcd(v[k],v[n-k])) ); v  \\ Joerg Arndt, Jun 30 2013

from math import gcd
A007464_list = [1, 1]
for n in range(1,10**3):
    A007464_list.append(sum(gcd(A007464_list[i],A007464_list[n-i]) for i in range(n+1)))
# Chai Wah Wu, Dec 26 2014

A007466 Exponential-convolution of natural numbers with themselves.

Original entry on oeis.org

1, 4, 14, 44, 128, 352, 928, 2368, 5888, 14336, 34304, 80896, 188416, 434176, 991232, 2244608, 5046272, 11272192, 25034752, 55312384, 121634816, 266338304, 580911104, 1262485504, 2734686208, 5905580032, 12717129728

Offset: 1

N. J. A. Sloane

Define a triangle T by T(n,1) = n*(n-1)+1 and T(r,c) = T(r,c-1) + T(r-1,c-1), then a(n) = T(n,n). - J. M. Bergot, Mar 03 2013
From David Callan, Jul 11 2014: (Start)
With offset 0, a(n) is the number of 2 X n 0-1 matrices that do not contain
  1 1    0 0
  0 0 or 1 1, as a 2 X 2 submatrix,
See Ju and Seo link, Theorem 3.2. (End)
a(n) is the sum of all ways of adding the k-tuples of the terms in the (n-1)-st row of Pascal's triangle A007318. For n=4 take row 3 of A007318: 1,3,3,1, giving (1)+(3)+(3)+(1)=8; (1+3)+(3+3)+(3+1)=14; (1+3+3)+(3+3+1)=14; (1+3+3+1)=8. The sum of these four terms is 8+14+14+8=44. - J. M. Bergot, Jun 17 2017
Binomial transform of A002061. - Jules Beauchamp, Jan 04 2022
a(n+1) is the number of strings of length n defined on {0,1,2,3} that contain at most one 2, at most one 3, and have no restriction on the number of 0s and 1s. For example, for n=2, a(3)=14 since from the 16 strings of length 2 we exclude 22 and 33. - Enrique Navarrete, May 03 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of 0/1-matrices avoiding some 2x2 matrices, arXiv:1107.1299 [math.CO], 2011.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A002061, A007318, A141611, A228643.

a007466 n = a228643 n n  -- Reinhard Zumkeller, Aug 29 2013

[2^(n-1)*(n+(n-1)*(n-2)/4) : n in [1..30]]; // Wesley Ivan Hurt, Jul 11 2014

A007466:=n->2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2): seq(A007466(n), n=1..30);

Table[2^(n-1)*(n + (n-1)*(n-2)/4), {n, 30}] (* Wesley Ivan Hurt, Jul 11 2014 *)

def A007466(n): return 2^(n-3)*(n^2+n+2)
[A007466(n) for n in range(1,31)] # G. C. Greubel, Sep 22 2024

E.g.f.: (Sum_{n >= 1} n*x^(n-1)/(n-1)!)^2.
a(n) = 2^(n-1)*n + 2^(n-3)*(n-1)*(n-2).
a(n) = Sum_{k=0..(n+2)} C(n+2, k) * floor(k/2)^2. - Paul Barry, Mar 06 2003
E.g.f.: (1+x)^2*exp(2*x). - Vladeta Jovovic, Sep 09 2003
G.f.: x*(1 - 2*x + 2*x^2)/(1-2*x)^3. - Vladimir Kruchinin, Sep 28 2011
E.g.f.: U(0) where U(k)= 1 + 2*x/( 1 - x/(2 + x - 4/( 2 + x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
a(n) = A228643(n, n). - Reinhard Zumkeller, Aug 29 2013
a(n) = Sum_{k=0..n-1} A141611(n-1, k). - G. C. Greubel, Sep 22 2024

A007550 Natural numbers exponentiated twice.

Original entry on oeis.org

1, 4, 20, 127, 967, 8549, 85829, 962308, 11895252, 160475855, 2343491207, 36795832297, 617662302441, 11031160457672, 208736299803440, 4169680371133507, 87648971646028515, 1933298000313801349, 44633323736412392093, 1076069422794010119112

Offset: 1

N. J. A. Sloane

The subsequence of primes (for n = 4, 5, 7) begins: 127, 967, 85829. The subsequence of semiprimes (for n = 2, 6) begins: 4, 8549. - Jonathan Vos Post, Feb 09 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..200
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 166
N. J. A. Sloane, Transforms

exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: a:= exptr(exptr(n->n)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

a[n_] := Sum[k^(n-k)*Binomial[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 11 2014, after Olivier Gérard *)

E.g.f.: exp(G(x) - 1) - 1, where G(x) = exp(x*exp(x)) = e.g.f. for A000248; clarified by Ilya Gutkovskiy, Jun 25 2018

a(n) = sum( k^(n - k) binomial(n,k) bell(k), k = 0..n ). - Olivier Gérard, Oct 24 2007

cp's OEIS Frontend

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A279215 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k+1)/6).

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