A002999 -id:A002999 - cp's OEIS frontend

A003013 E.g.f. 1 + xexp(x) + x^2exp(2*x).

1, 1, 4, 15, 52, 165, 486, 1351, 3592, 9225, 23050, 56331, 135180, 319501, 745486, 1720335, 3932176, 8912913, 20054034, 44826643, 99614740, 220200981, 484442134, 1061158935, 2315255832, 5033164825

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-25,38,-28,8).

Cf. A048482, A001787, A005183.

Cf. A002999, A001815.

With[{nn=30},CoefficientList[Series[1+x Exp[x]+x^2 Exp[2x],{x,0,nn}],x] Range[0,nn]!] (* or *) Join[{1},LinearRecurrence[{8,-25,38,-28,8},{1,4,15,52,165},30]] (* Harvey P. Dale, Nov 01 2011 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 8,-28,38,-25,8]^n*[1;1;4;15;52])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

From Ralf Stephan, Sep 02 2003: (Start)
a(0) = 1, a(n) = (n^2 - n)*2^n/4 + n.
a(n) = A002999(n) - n = A001815(n) + n. (End)
O.g.f.: 1+x*(-1+4*x-8*x^2+6*x^3) / ( (x-1)^2*(2*x-1)^3 ). - R. J. Mathar, Mar 22 2011
a(n) = 8*a(n-1) - 25*a(n-2) + 38*a(n-3) - 28*a(n-4) + 8*a(n-5); a(0)=1, a(1)=1, a(2)=4, a(3)=15, a(4)=52, a(5)=165. - Harvey P. Dale, Nov 01 2011

A048482 a(n) = T(n,n), array T given by A048472.

Original entry on oeis.org

1, 3, 13, 49, 161, 481, 1345, 3585, 9217, 23041, 56321, 135169, 319489, 745473, 1720321, 3932161, 8912897, 20054017, 44826625, 99614721, 220200961, 484442113, 1061158913, 2315255809, 5033164801, 10905190401, 23555211265, 50734301185, 108984795137

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A002999, A003013, A001787, A005183, A001788.

[(n^2+n)*2^(n-1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

LinearRecurrence[{7,-18,20,-8},{1,3,13,49},30] (* Harvey P. Dale, Feb 02 2015 *)

Vec(-(8*x^3-10*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Nov 26 2014

a(n) = 2 * A001788(n) + 1.
a(n) = (n^2+n)*2^(n-1) + 1. - Ralf Stephan, Sep 02 2003
G.f.: -(8*x^3-10*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Nov 26 2014
a(0)=1, a(1)=3, a(2)=13, a(3)=49, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Feb 02 2015

A377574 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^2.

Original entry on oeis.org

1, 2, 14, 150, 2264, 44370, 1073772, 30998954, 1041094448, 39909978594, 1720526113460, 82422717484602, 4345035540566184, 250012958308399442, 15594180423126432428, 1048169467357831893930, 75535629221800163853152, 5810132660615400890909634, 475146028302302130377698404

Offset: 0

Seiichi Manyama, Nov 02 2024

nonn

Cf. A006153, A377575, A377576.

Cf. A002999, A295238, A377553.

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A295238.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+2,k)/( (k+1)*(n-k)! ).
a(n) ~ 2^(5/2) * sqrt(1 + LambertW(1/4)) * n^(n-1) / (LambertW(1/4)^n * exp(n)). - Vaclav Kotesovec, Nov 02 2024

A386254 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times.

Original entry on oeis.org

1, 1, 2, 6, 18, 60, 240, 1085, 5012, 23730, 121440, 685707, 4144668, 25614589, 159141892, 1012740885, 6805631232, 48872707006, 369227821608, 2853779791619, 22131042288980, 172055270717463, 1362017827326860, 11208504802237327, 96939147303239304, 875473007351905045

Offset: 0

John Tyler Rascoe, Jul 16 2025

			a(3) = 6 counts: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2), (3,3,3).

Alois P. Heinz, Table of n, a(n) for n = 0..674

Cf. A002999, A005183, A007837, A032299, A347006, A386255.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/j!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 17 2025

terms=26; CoefficientList[Series[Product[1+Sum[x^j/j!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)

D_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/i!))))}

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / j!).

A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.

Original entry on oeis.org

1, 1, 4, 15, 64, 325, 1776, 11179, 72640, 489969, 3435580, 26495491, 221599104, 1893705697, 16145571820, 138299146665, 1241234863936, 12033569772769, 124055067568788, 1303750295285563, 13577876900409280, 139418829477000801, 1441311794301705964, 15537427948684769425

Offset: 0

John Tyler Rascoe, Jul 16 2025

			a(3) = 15 counts: (1#,1,1), (1,1#,1), (1,1,1#), (1#,2#,2), (1#,2,2#), (2#,1#,2), (2,1#,2#), (2#,2,1#), (2,2#,1#), (2#,2,2), (2,2#,2), (2,2,2#), (3#,3,3), (3,3#,3), (3,3,3#) where # denotes a mark.

Alois P. Heinz, Table of n, a(n) for n = 0..650

Cf. A002999, A005183, A007837, A032299, A347006, A386254.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-j, min(n-j, i-1))/(j-1)!, j=i..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 17 2025

terms=24; CoefficientList[Series[Product[1+Sum[x^j/(j-1)!, {j,k,terms}],{k,terms}],{x,0,terms-1}],x]Range[0,terms-1]! (* Stefano Spezia, Jul 17 2025 *)

E_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(prod(k=1,N, 1 + sum(i=k,N, x^i/((i-1)!)))))}

E.g.f.: Product_{k>=1} (1 + Sum_{j>=k} x^j / (j-1)!).

A377582 Expansion of e.g.f. (1 + x * exp(x))^3.

Original entry on oeis.org

1, 3, 12, 51, 228, 1035, 4698, 21063, 92424, 395091, 1643790, 6664383, 26387100, 102286587, 389125506, 1455994935, 5368721808, 19541252259, 70312410774, 250408115823, 883617559140, 3092276105163, 10740749281482, 37053754521831, 127037475064728, 433073722098675

Offset: 0

Seiichi Manyama, Nov 02 2024

Cf. A002999, A377583.
Cf. A001815, A052791.
Cf. A377530, A377575.

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3, k)/(n-k)!);

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3,k)/(n-k)!.

G.f.: (1-17*x+127*x^2-542*x^3+1453*x^4-2543*x^5+2973*x^6-2478*x^7+1626*x^8-648*x^9) / ((1-x)^2*(1-2*x)^3*(1-3*x)^4).

A377583 Expansion of e.g.f. (1 + x * exp(x))^4.

Original entry on oeis.org

1, 4, 20, 108, 616, 3620, 21624, 129892, 778208, 4621572, 27080680, 156080804, 883304976, 4905620356, 26743018904, 143219056740, 754280089024, 3911369843204, 19995029207496, 100885122939172, 502952669726960, 2480084192804484, 12107351426245240, 58565261434872548

Offset: 0

Seiichi Manyama, Nov 02 2024

Cf. A002999, A377582.

Cf. A377399, A377576.

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4, k)/(n-k)!);

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4,k)/(n-k)!.

G.f.: (1 - 36*x + 595*x^2 - 5970*x^3 + 40543*x^4 - 196752*x^5 + 702365*x^6 - 1871250*x^7 + 3740456*x^8 - 5614440*x^9 + 6362360*x^10 - 5588880*x^11 + 3979680*x^12 - 2196672*x^13 + 663552*x^14) / ((1-x)^2*(1-2*x)^3*(1-3*x)^4*(1-4*x)^5).

cp's OEIS Frontend

A003013 E.g.f. 1 + x*exp(x) + x^2*exp(2*x).

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PARI

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A048482 a(n) = T(n,n), array T given by A048472.

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Magma

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A377574 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^2.

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A386254 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times.

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Maple

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A386255 Number of words of length n over an infinite alphabet such that for any letter k appearing within a word the letter k appears at least k times and exactly one of each kind of letter is marked.

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Maple

Mathematica

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A377582 Expansion of e.g.f. (1 + x * exp(x))^3.

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PARI

Formula

A377583 Expansion of e.g.f. (1 + x * exp(x))^4.

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A003013 E.g.f. 1 + xexp(x) + x^2exp(2*x).