cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A326324 a(n) = A_{5}(n) where A_{m}(x) are the Eulerian polynomials as defined in A326323.

Original entry on oeis.org

1, 1, 6, 46, 456, 5656, 84336, 1467376, 29175936, 652606336, 16219458816, 443419545856, 13224580002816, 427278468668416, 14867050125981696, 554245056343668736, 22039796215883268096, 931198483176870608896, 41658202699736550014976, 1967160945260218035798016
Offset: 0

Views

Author

Peter Luschny, Jun 27 2019

Keywords

Comments

See A326323 for the more general formulas.

Crossrefs

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326323.

Programs

  • Maple
    seq(add(combinat:-eulerian1(n,k)*5^k, k=0..n), n=0..20);
# Alternative:
egf := 4/(5 - exp(4*x)): ser := series(egf, x, 21):
seq(k!*coeff(ser, x, k), k=0..20);
  • Mathematica
    a[1] := 1; a[n_] := 4^(n + 1)/5 HurwitzLerchPhi[1/5, -n, 0];
Table[a[n], {n, 0, 20}]
(* Alternative: *)
s[n_] := Sum[StirlingS2[n, j] 4^(n - j) j!, {j, 0, n}];
Table[s[n], {n, 0, 20}]

Formula

a(n) ~ n!/5 * (4/log(5))^(n+1). - Vaclav Kotesovec, Aug 09 2021
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k). - Ilya Gutkovskiy, Feb 04 2022

Extensions

Corrected after notice from Jean-François Alcover by Peter Luschny, Jul 13 2019

A336951 E.g.f.: 1 / (1 - x * exp(3*x)).

Original entry on oeis.org

1, 1, 8, 69, 780, 11145, 191178, 3823785, 87406056, 2247785073, 64228084110, 2018771719569, 69221032558956, 2571290056399545, 102860527370221026, 4408690840306136505, 201557641172689004112, 9790792086366911655009, 503570143277542340304534
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2020

Keywords

Links

Crossrefs

Column k=3 of A351790.
Cf. A006153, A027471, A235328, A255927, A328182, A336948, A336950, A336952.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[1/(1 - x Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! Sum[(3 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
  • PARI
    seq(n)={ Vec(serlaplace(1 / (1 - x*exp(3*x + O(x^n))))) } \\ Andrew Howroyd, Aug 08 2020

Formula

a(n) = n! * Sum_{k=0..n} (3 * (n-k))^k / k!.
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 3^(k-1) * a(n-k).
a(n) ~ n! * (3/LambertW(3))^n / (1 + LambertW(3)). - Vaclav Kotesovec, Aug 09 2021

A326323 A(n, k) = A_{n}(k) where A_{n}(x) are the Eulerian polynomials, square array read by ascending antidiagonals, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 13, 24, 1, 1, 1, 5, 22, 75, 120, 1, 1, 1, 6, 33, 160, 541, 720, 1, 1, 1, 7, 46, 285, 1456, 4683, 5040, 1, 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1, 1, 1, 9, 78, 679, 5656, 40005, 202672, 545835, 362880, 1
Offset: 0

Views

Author

Peter Luschny, Jun 27 2019

Keywords

Examples

			Array starts:
  k=0: 1, 1, 1,  1,    1,     1,      1,        1,         1, ... [A000012]
  k=1: 1, 1, 2,  6,   24,   120,    720,     5040,     40320, ... [A000142]
  k=2: 1, 1, 3, 13,   75,   541,   4683,    47293,    545835, ... [A000670]
  k=3: 1, 1, 4, 22,  160,  1456,  15904,   202672,   2951680, ... [A122704]
  k=4: 1, 1, 5, 33,  285,  3081,  40005,   606033,  10491885, ... [A255927]
  k=5: 1, 1, 6, 46,  456,  5656,  84336,  1467376,  29175936, ... [A326324]
  k=6: 1, 1, 7, 61,  679,  9445, 158095,  3088765,  68958295, ... [A384525]
  k=7: 1, 1, 8, 78,  960, 14736, 272448,  5881968, 145105920, ... [A384514]
  k=8: 1, 1, 9, 97, 1305, 21841, 440649, 10386817, 279768825, ...
Seen as a triangle:
  [0], 1
  [1], 1, 1
  [2], 1, 1, 1
  [3], 1, 1, 2,  1
  [4], 1, 1, 3,  6,   1
  [5], 1, 1, 4, 13,  24,    1
  [6], 1, 1, 5, 22,  75,  120,     1
  [7], 1, 1, 6, 33, 160,  541,   720,     1
  [8], 1, 1, 7, 46, 285, 1456,  4683,  5040,     1
  [9], 1, 1, 8, 61, 456, 3081, 15904, 47293, 40320, 1

Links

Crossrefs

Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A384525, A384514.

Programs

  • Maple
    A := (n, k) -> add(combinat:-eulerian1(k, j)*n^j, j=0..k):
seq(seq(A(n-k, k), k=0..n), n=0..10);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n  - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(k!*coeff(ser(n), x, k), k=0..9) od;
  • Mathematica
    a[n_, 0] := 1; a[n_, 1] := n!;
a[n_, k_] := (k - 1)^(n + 1)/k HurwitzLerchPhi[1/k, -n, 0];
(* Alternative: *) a[n_, k_] := Sum[StirlingS2[n, j] (k - 1)^(n - j) j!, {j, 0, n}];
Table[Print[Table[a[n, k], {n, 0, 10}]], {k, 0, 8}]

Formula

A(n, k) = Sum_{j=0..k} a(k, j)*n^j where a(k, j) are the Eulerian numbers.
E.g.f.: (n - 1)/(n - exp((n-1)*x)) for n = 0 and n >= 2, 1/(1 - x) if n = 1.
A(n, 0) = 1; A(n, 1) = n!.
A(n, k) = (k - 1)^(n + 1)/k HurwitzLerchPhi(1/k, -n, 0) for k >= 2.
A(n, k) = Sum_{j=0..n} j! * Stirling2(n, j) * (k - 1)^(n - j) for k >= 2.

A355111 Expansion of e.g.f. 3 / (4 - 3*x - exp(3*x)).

Original entry on oeis.org

1, 2, 11, 93, 1041, 14541, 243747, 4767183, 106556373, 2679469065, 74864397015, 2300883358995, 77144051804409, 2802027511061325, 109604157405491691, 4593512301562215783, 205348466229473678301, 9753645833118762303249, 490530576727430107027839, 26040317900991310393061499
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A003575, A006155, A032033, A255927, A343673, A355110, A355112, A355113, A355114.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[3/(4 - 3 x - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 3^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(4))) * ((4 - LambertW(exp(4)))/3)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A352069 Expansion of e.g.f. 1 / (1 + log(1 - 3*x) / 3).

Original entry on oeis.org

1, 1, 5, 42, 492, 7374, 134478, 2887128, 71281656, 1988802720, 61860849552, 2121993490176, 79566300371952, 3237181141173264, 142019158472311248, 6682603650677875584, 335698708873243355136, 17930674324049810882688, 1014685181110897126616448, 60641642160287342580586752
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 02 2022

Keywords

Crossrefs

Cf. A007840, A032031, A087674, A227917, A255927, A352071.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 + Log[1 - 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-3)^(n - k), {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-3*x)/3))) \\ Michel Marcus, Mar 02 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-3)^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * 3^(k-1) * a(n-k).
a(n) ~ n! * 3^(n+1) * exp(3*n) / (exp(3) - 1)^(n+1). - Vaclav Kotesovec, Mar 03 2022

A351757 G.f. A(x) satisfies: A(x) = 1 + x * A(x/(1 - 3*x)) / (1 - 3*x)^2.

Original entry on oeis.org

1, 1, 7, 43, 289, 2239, 19699, 192025, 2042971, 23520715, 291099349, 3849621019, 54110928355, 804827487493, 12619011606775, 207885167529523, 3587864566792753, 64705561315720135, 1216574535057705979, 23797327657083197113, 483390249416359706995
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 18 2022

Keywords

Crossrefs

Cf. A004212, A040027, A255927, A351756.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1 + x A[x/(1 - 3 x)]/(1 - 3 x)^2 + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k - 1] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k-1) * 3^(k-1) * a(n-k).

A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

Original entry on oeis.org

1, 1, 4, 33, 456, 9445, 272448, 10386817, 503758720, 30202999821, 2189000524800, 188349613075393, 18954958449853440, 2203304642871358741, 292675996808408743936, 44022321302156791898625, 7438113993194856900034560, 1401876939543892434209075581
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 24 2020

Keywords

Links

Crossrefs

Cf. A000670, A063170, A086914, A094420, A122704, A122778, A229234, A255927, A301419, A326323, A326324.

Programs

  • Mathematica
    Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(n - k), {k, 0, n}], {n, 1, 17}]]
Table[SeriesCoefficient[Sum[k! x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n^(n + 1) PolyLog[-n, 1/(n + 1)]/(n + 1), {n, 1, 17}]]
  • PARI
    a(n) = sum(k=0, n, stirling(n, k, 2)*k!*n^(n-k)); \\ Michel Marcus, Jan 24 2020

Formula

a(n) = [x^n] Sum_{k>=0} k! * x^k / Product_{j=1..k} (1 - n*j*x).
a(n) = n! * [x^n] n / (1 + n - exp(n*x)) for n > 0.
a(n) = n^(n + 1) * Sum_{k>=1} k^n / (n + 1)^(k + 1) for n > 0.
a(n) ~ n! * n^(n+1) / ((n+1) * log(n+1)^(n+1)). - Vaclav Kotesovec, Jun 06 2022

A354750 Expansion of e.g.f. 1 / (1 - log(1 + 3*x) / 3).

Original entry on oeis.org

1, 1, -1, 6, -48, 534, -7542, 129240, -2603736, 60292512, -1577546928, 46021512096, -1480976147664, 52110720451152, -1990258155061776, 81995762243700864, -3624527727510038784, 171109526616468957312, -8591991935936929932672, 457246520477143117555968
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2022

Keywords

Crossrefs

Cf. A006252, A087674, A255927, A335531, A352069, A354237, A354263, A354751.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - Log[1 + 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 3^(n - k), {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+3*x)/3))) \\ Michel Marcus, Jun 06 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 3^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-3)^(k-1) * a(n-k).

A340887 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * 3^(n-k-1) * a(k).

Original entry on oeis.org

1, 1, 7, 99, 2511, 99531, 5680125, 441226521, 44766049599, 5748319130283, 911271895816077, 174799606363478361, 39903413238125862309, 10690643656077551475921, 3321750648705212259711063, 1184831658624977151885176859, 480843465699932167142334581919
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 25 2021

Keywords

Crossrefs

Cf. A102221, A255927, A340886, A340888.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^2 3^(n - k - 1) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
nmax = 16; CoefficientList[Series[3/(4 - BesselI[0, 2 Sqrt[3 x]]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = 3 / (4 - BesselI(0,2*sqrt(3*x))).

A332700 A(n, k) = Sum_{j=0..n} j!*Stirling2(n, j)*(k-1)^(n-j), for n >= 0 and k >= 0, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 24, 13, 4, 1, 1, 1, 120, 75, 22, 5, 1, 1, 1, 720, 541, 160, 33, 6, 1, 1, 1, 5040, 4683, 1456, 285, 46, 7, 1, 1, 1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1, 1, 362880, 545835, 202672, 40005, 5656, 679, 78, 9, 1, 1
Offset: 0

Views

Author

Peter Luschny, Feb 28 2020

Keywords

Examples

			Array begins:
[0] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[1] 1, 1,       1,       1,        1,         1,         1, ...    A000012
[2] 1, 2,       3,       4,        5,         6,         7, ...    A000027
[3] 1, 6,       13,      22,       33,        46,        61, ...   A028872
[4] 1, 24,      75,      160,      285,       456,       679, ...
[5] 1, 120,     541,     1456,     3081,      5656,      9445, ...
[6] 1, 720,     4683,    15904,    40005,     84336,     158095, ...
[7] 1, 5040,    47293,   202672,   606033,    1467376,   3088765, ...
[8] 1, 40320,   545835,  2951680,  10491885,  29175936,  68958295, ...
[9] 1, 362880,  7087261, 48361216, 204343641, 652606336, 1731875605, ...
       A000142, A000670, A122704,  A255927,   A326324, ...
Seen as a triangle:
[0] [1]
[1] [1, 1]
[2] [1, 1,     1]
[3] [1, 2,     1,     1]
[4] [1, 6,     3,     1,     1]
[5] [1, 24,    13,    4,     1,    1]
[6] [1, 120,   75,    22,    5,    1,   1]
[7] [1, 720,   541,   160,   33,   6,   1,  1]
[8] [1, 5040,  4683,  1456,  285,  46,  7,  1, 1]
[9] [1, 40320, 47293, 15904, 3081, 456, 61, 8, 1, 1]

Links

Crossrefs

The matrix transpose of A326323.
Cf. A173018, A000012, A000142, A000670, A122704, A255927, A326324, A000027, A028872.

Programs

  • Maple
    # Prints array by row.
A := (n, k) -> add(combinat:-eulerian1(n, j)*k^j, j=0..n):
seq(print(seq(A(n,k), k=0..10)), n=0..8);
# Alternative:
egf := n -> `if`(n=1, 1/(1-x), (n-1)/(n - exp((n-1)*x))):
ser := n -> series(egf(n), x, 21):
for n from 0 to 6 do seq(n!*coeff(ser(k), x, n), k=0..9) od;
# Or:
A := (n, k) -> if k = 0 or n = 0 then 1 elif k = 1 then n! else
polylog(-n, 1/k)*(k-1)^(n+1)/k fi:
for n from 0 to 6 do seq(A(n, k), k=0..9) od;
  • Mathematica
    A332700[n_, k_] := n! + Sum[j! StirlingS2[n, j] (k-1)^(n-j), {j, n-1}];
Table[A332700[n-k, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 01 2024 *)
  • Sage
    def T(n, k):
    return sum(factorial(j)*stirling_number2(n, j)*(k-1)^(n-j) for j in range(n+1))
for n in range(8): print([T(n, k) for k in range(8)])

Formula

A(n, k) = Sum_{j=0..n} E(n, j)*k^j, where E(n, k) = A173018(n, k).
A(n, 1) = n!*[x^n] 1/(1-x).
A(n, k) = n!*[x^n] (k-1)/(k - exp((k-1)*x)) for k != 1.
A(n, k) = PolyLog(-n, 1/k)*(k-1)^(n+1)/k for k >= 2.
Showing 1-10 of 10 results.

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