A213236 -id:A213236 - cp's OEIS frontend

A361291 a(n) = ((2n + 1)^n - 1)/(2n).

1, 6, 57, 820, 16105, 402234, 12204241, 435984840, 17927094321, 833994048910, 43309534450633, 2483526865641276, 155867505885345241, 10627079738421409410, 782175399728156197665, 61812037545704964935440, 5220088150634922700769761, 469168161404536131943150998

Offset: 1

Stefano Spezia, Mar 12 2023

This sequence is of the form (k^n - 1)/(k - 1) with k = 2*n + 1. See crossrefs in A218722 for other sequences of the same form.

Cf. A005408, A019762 (2*e), A051129, A218722.

Cf. A000169, A000312, A038057, A052746, A062971, A213236.

Mathematica
```
Table[((2n+1)^n-1)/(2n),{n,20}]
```

def A361291(n): return (((n<<1)+1)**n-1)//(n<<1) # Chai Wah Wu, Mar 14 2023

a(n) = Sum_{i=0..n-1} A005408(n)^i.
a(n) = n! * [x^n] exp(x)*(exp(2*n*x) - 1)/(2*n).
a(n) = n! * [x^n] exp((n+1)*x)*sinh(n*x)/n.
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e.
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = 2*e.

A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).

Original entry on oeis.org

1, 0, 2, 0, 3, 12, 0, 4, 60, 120, 0, 5, 210, 1260, 1680, 0, 6, 630, 8400, 30240, 30240, 0, 7, 1736, 45360, 327600, 831600, 665280, 0, 8, 4536, 216720, 2772000, 13305600, 25945920, 17297280, 0, 9, 11430, 956340, 20207880, 162162000, 575134560, 908107200, 518918400

Offset: 0

Peter Luschny, Jan 10 2024

			Triangle starts:
  [0] [1]
  [1] [0, 2]
  [2] [0, 3,   12]
  [3] [0, 4,   60,    120]
  [4] [0, 5,  210,   1260,    1680]
  [5] [0, 6,  630,   8400,   30240,    30240]
  [6] [0, 7, 1736,  45360,  327600,   831600,   665280]
  [7] [0, 8, 4536, 216720, 2772000, 13305600, 25945920, 17297280]

Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Cf. A124320 (rising factorial), A048993(Stirling2), A053492 (row sums), A213236 (alternating row sums), A001813 (main diagonal), A368583.

def Trow(n): return [rising_factorial(n+1, k)*stirling_number2(n, k) for k in range(n+1)]
for n in range(7): print(Trow(n))

A383130 Coefficients of the linear terms in the continued fraction representation of the product logarithm.

Original entry on oeis.org

1, 1, 1, 5, 17, 133, 1927, 13582711, 92612482895, 10402118970990527, 59203666396198716260449, 83631044830029201279016528831, 1149522186344339904123210420373026673, 458029700061597358458976211208014885543904637441, 203695852839150317577316770934832249000714992664672874100151

Offset: 1

Jacob DeMoss, Jun 17 2025

nonn

The continued fraction only produces values for the principal branch of the product logarithm.

			LambertW(x) = x/(1 + x/(1 + x/(2 + 5*x/(3 + 17*x/(10 + 133*x/(17 + 1927*x/(190 + ... ))))))).

Cristina B. Corcino, Roberto B. Corcino, and István Mező, Continued fraction expansions for the Lambert W function, Aequat. Math. 93, 485-498 (2019)

Cf. A213236 (e.g.f. of LambertW).

ClearAll[cf, x];
cf[ O[x]] = {};
cf[ a0_ + O[x]] := {a0};
cf[ ps_] := Module[ {a0, a1, u, v},
  a0 = SeriesCoefficient[ ps, {x, 0, 0}];
  a1 = SeriesCoefficient[ ps, {x, 0, 1}];
  u = Numerator[a1]; v = Denominator[a1];
  Join[ If[ a0==0, {}, {a0}],
     Prepend[cf[ u*x/(ps-a0) - v], {u,v}]]];
(* Lambert W function W_0(x) up to O(x)^(M+1) *)
M = 10; W0 = Sum[ x^n*(-n)^(n-1)/n!, {n, 1, M}] + x*O[x]^M;
cf[W0] //InputForm
(* {{1, 1}, {1, 1}, {1, 2}, {5, 3}, {17, 10}, {133, 17},
 {1927, 190}, {13582711, 94423}, {92612482895, 1597966},
 {10402118970990527, 8773814169}} *)
(* Note: Change M to the number of terms to be generated *)

More terms from Alois P. Heinz, Jun 17 2025

cp's OEIS Frontend

A361291 a(n) = ((2n + 1)^n - 1)/(2n).

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A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).

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A383130 Coefficients of the linear terms in the continued fraction representation of the product logarithm.

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cp's OEIS Frontend

A361291 a(n) = ((2*n + 1)^n - 1)/(2*n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Formula

A368584 Table read by rows: T(n, k) = A124320(n + 1, k) * A048993(n, k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

SageMath

A383130 Coefficients of the linear terms in the continued fraction representation of the product logarithm.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A361291 a(n) = ((2n + 1)^n - 1)/(2n).