A072034 -id:A072034 - cp's OEIS frontend

A245260 Decimal expansion of the root of the equation r*log(r/(1-r))=1.

7, 8, 2, 1, 8, 8, 2, 9, 4, 2, 8, 0, 1, 9, 9, 9, 0, 1, 2, 2, 0, 2, 9, 7, 0, 7, 5, 9, 2, 6, 7, 4, 4, 7, 8, 0, 1, 8, 1, 9, 0, 8, 4, 0, 3, 9, 6, 6, 2, 9, 9, 5, 1, 6, 8, 7, 0, 9, 6, 8, 3, 3, 2, 3, 9, 5, 6, 9, 1, 6, 9, 9, 4, 1, 2, 4, 6, 7, 4, 6, 7, 1, 9, 5, 3, 8, 2, 3, 9, 2, 9, 0, 6, 6, 7, 3, 2, 5, 1, 3, 6, 6, 7, 5, 8, 5

Offset: 0

Vaclav Kotesovec, Jul 15 2014

			0.78218829428019990122029707592674478018190840396629951687...

Cf. A072034, A220359, A237421, A245259.

Maple
```
evalf(solve(r*log(r/(1-r))=1), 100)
```

RealDigits[r/.FindRoot[r*Log[r/(1-r)]==1, {r, 3/4}, WorkingPrecision->250], 10, 200][[1]]
RealDigits[1/(1+LambertW[E^(-1)]), 10, 200][[1]]

Equals 1/(1+LambertW(exp(-1))).

A349706 Square array T(n,k) = Sum_{j=0..k} binomial(k,j) * j^n for n and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 1, 4, 8, 0, 1, 6, 12, 16, 0, 1, 10, 24, 32, 32, 0, 1, 18, 54, 80, 80, 64, 0, 1, 34, 132, 224, 240, 192, 128, 0, 1, 66, 342, 680, 800, 672, 448, 256, 0, 1, 130, 924, 2192, 2880, 2592, 1792, 1024, 512, 0, 1, 258, 2574, 7400, 11000, 10752, 7840, 4608, 2304, 1024

Offset: 0

Michel Marcus, Nov 26 2021

			Square array begins:
  1 2  4   8   16    32
  0 1  4  12   32    80
  0 1  6  24   80   240
  0 1 10  54  224   800
  0 1 18 132  680  2880
  0 1 34 342 2192 11000

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Renate Golombek, Aufgabe 1088, El. Math., 49 (1994), 126-127.
Simsek Yilmaz, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Applicable Analysis and Discrete Mathematics 2018 Volume 12, Issue 1, Pages: 1-35. See B(n,k).

Rows n=0..7 give A000079, A001787, A001788, A058645, A058649, A059338, A056468, A084641.
Main diagonal gives A072034.
Cf. A209849.

T[n_, k_] := Sum[Binomial[k, j] * If[j == n == 0, 1, j^n], {j, 0, k}]; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 26 2021 *)

T(n,k) = sum(j=0, k, binomial(k,j)*j^n);

A362604 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^2))).

Original entry on oeis.org

1, 1, 4, 33, 352, 4805, 80256, 1582693, 36001792, 927974601, 26729943040, 850921057481, 29666297020416, 1124166449205709, 46005243970846720, 2022121401647311245, 95008417631810093056, 4751844218849365365137, 252063937292253895065600

Offset: 0

Seiichi Manyama, Apr 30 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A072034, A362699.

Cf. A216688, A356834, A362700, A362702.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x*exp(x^2)))))

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^(n-k) / (k! * (n-2*k)!).

A362701 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^3/6))).

Original entry on oeis.org

1, 1, 4, 27, 260, 3205, 48276, 859453, 17656696, 411139233, 10700380520, 307819026031, 9698757574716, 332170854765373, 12286858280098780, 488160559069250985, 20732661511284180656, 937357753835195873857, 44948438093966732331984

Offset: 0

Seiichi Manyama, Apr 30 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A072034, A362700.

Cf. A362705.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x*exp(x^3/6)))))

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)^(n-2*k) / (6^k * k! * (n-3*k)!).

A362704 Expansion of e.g.f. 1/(1 + LambertW(-x^2/2 * exp(x))).

Original entry on oeis.org

1, 0, 1, 3, 18, 130, 1140, 11886, 142408, 1934640, 29357100, 492249340, 9038206056, 180352513848, 3886286296984, 89937276717120, 2224716791224320, 58577968147130176, 1635780290409117648, 48286974141713673072, 1502385897082471446880

Offset: 0

Seiichi Manyama, Apr 30 2023

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A072034, A362705.

Cf. A362350.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^2/2*exp(x)))))

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-k) / (2^k * k! * (n-2*k)!).

A366151 a(n) = T(n, 3), where T(n, k) = Sum_{i=0..n} i^k * binomial(n, i) * (1/2)^(n-k).

Original entry on oeis.org

0, 4, 20, 54, 112, 200, 324, 490, 704, 972, 1300, 1694, 2160, 2704, 3332, 4050, 4864, 5780, 6804, 7942, 9200, 10584, 12100, 13754, 15552, 17500, 19604, 21870, 24304, 26912, 29700, 32674, 35840, 39204, 42772, 46550

Offset: 0

Peter Luschny, Oct 27 2023

nonn

A mean of binomials as might occur as the Expectation of random variables.

T(n, 0) = A000012; T(n, 1) = A001477; T(n, 2) = A002378; T(n, 3) = this sequence.

T(1, n) = A011782; T(2, n) = A063376(n) (with offset 0); T(n, n) = A072034(n).

a := n -> n^2*(n + 3): seq(a(n), n = 0..35);

a(n) = n^2*(n + 3).
a(n) = [x^n] (2*x*(2 + 2*x - x^2))/(x - 1)^4.
a(n) = n! * [x^n] exp(x)*(x^3 + 6*x^2 + 4*x).

cp's OEIS Frontend

A245260 Decimal expansion of the root of the equation r*log(r/(1-r))=1.

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A349706 Square array T(n,k) = Sum_{j=0..k} binomial(k,j) * j^n for n and k >= 0, read by ascending antidiagonals.

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A362604 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^2))).

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A362701 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^3/6))).

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A362704 Expansion of e.g.f. 1/(1 + LambertW(-x^2/2 * exp(x))).

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A366151 a(n) = T(n, 3), where T(n, k) = Sum_{i=0..n} i^k * binomial(n, i) * (1/2)^(n-k).

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