A263576 -id:A263576 - cp's OEIS frontend

A263575 Stirling transform of Lucas numbers (A000032).

2, 1, 4, 14, 53, 227, 1092, 5791, 33350, 206511, 1365563, 9590847, 71216713, 556861216, 4569168866, 39222394456, 351304769679, 3275433717440, 31723522878974, 318571978752719, 3311400814816987, 35573458376435132, 394404160256111139, 4507130777468928696

Offset: 0

Vladimir Reshetnikov, Oct 21 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..563
Eric Weisstein's MathWorld, Lucas Number.
Eric Weisstein's MathWorld, Stirling Transform.
Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A000032, A213593, A005248, A061084, A263576.

Table[Sum[LucasL[k] StirlingS2[n, k], {k, 0, n}], {n, 0, 23}]
Table[Simplify[BellB[n, GoldenRatio] + BellB[n, 1 - GoldenRatio]], {n, 0, 23}]

a(n) = Sum_{k=0..n} A000032(k)*Stirling2(n,k).
Let phi = (1+sqrt(5))/2.
a(n) = B_n(phi)+B_n(1-phi), where B_n(x) is n-th Bell polynomial.
2*B_n(phi) = a(n) + A263576*sqrt(5).
E.g.f.: exp((exp(x)-1)*phi)+exp((exp(x)-1)*(1-phi)).
Sum_{k=0..n} a(k)*Stirling1(n,k) = A000032(n).
G.f.: Sum_{j>=0} Lucas(j)*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019

A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).

Original entry on oeis.org

0, 1, 3, 19, 159, 1651, 20583, 299419, 4977759, 93097891, 1934655063, 44224195819, 1102820674959, 29792843865331, 866769668577543, 27018340680076219, 898343366411181759, 31736205208791131971, 1187110673532381604023, 46871464129796857140619, 1948059531745350527058159

Offset: 0

Ilya Gutkovskiy, Oct 11 2018

nonn

From Peter Bala, Aug 19 2025: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, ...] with an apparent period of 6 = phi(9) beginning at n = 2. Cf. A004123.(End)

Seiichi Manyama, Table of n, a(n) for n = 0..401

Cf. A000045, A000556, A000557, A005443, A005445, A048993, A263576.

seq(n!*coeff(series((exp(x) - 1)/(exp(x) - exp(2*x) + 1), x=0, 22), x, n), n=0..21); # Paolo P. Lava, Jan 09 2019

nmax = 20; CoefficientList[Series[(Exp[x] - 1)/(Exp[x] - Exp[2 x] + 1), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] Fibonacci[k] k!, {k, 0, n}], {n, 0, 20}]

E.g.f.: (1 + sinh(x) - cosh(x))/(1 - 2*sinh(x)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*Fibonacci(k)*k!.
a(n) ~ n! / (sqrt(5) * phi^2 * (log(phi))^(n+1)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 12 2018

A323620 Expansion of e.g.f. 2sqrt(1 + x)sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).

Original entry on oeis.org

0, 1, 0, 1, -4, 19, -108, 719, -5496, 47465, -457160, 4858865, -56490060, 713165035, -9715762980, 142069257055, -2219386098160, 36889108220305, -650018185589520, 12103669982341025, -237476572759473300, 4896758300881695875, -105866710959427454300, 2394660132226522508975

Offset: 0

Ilya Gutkovskiy, Jan 20 2019

sign

G. C. Greubel, Table of n, a(n) for n = 0..449

Cf. A000045, A048994, A178840, A213593, A263576, A265165.

[(&+[StirlingFirst(n,k)*Fibonacci(k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019

FullSimplify[nmax = 23; CoefficientList[Series[2 Sqrt[1 + x] Sinh[Sqrt[5] Log[1 + x]/2]/Sqrt[5], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS1[n, k] Fibonacci[k], {k, 0, n}], {n, 0, 23}]

{a(n) = sum(k=0,n, stirling(n,k,1)*fibonacci(k))};
vector(25, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019

[sum((-1)^(k+n)*stirling_number1(n,k)*fibonacci(k) for k in (0..n)) for n in (0..25)] # G. C. Greubel, Feb 07 2019

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000045(k).
From Vaclav Kotesovec, Jan 21 2019: (Start)
a(n) = -(-1)^n * cos(sqrt(5)*Pi/2) * (Gamma((3 + sqrt(5))/2) * Gamma(n - (1 + sqrt(5))/2) - Gamma((3 - sqrt(5))/2) * Gamma(n + (sqrt(5) - 1)/2)) / (Pi*sqrt(5)).
a(n) ~ -(-1)^n * n! / (sqrt(5) * Gamma((sqrt(5)-1)/2) * n^((3 - sqrt(5))/2)).
a(n) = -2*(n-2)*a(n-1) - (n^2 - 5*n + 5)*a(n-2). (End)

A323631 Stirling transform of Pell numbers (A000129).

Original entry on oeis.org

0, 1, 3, 12, 57, 305, 1798, 11531, 79707, 589426, 4634471, 38547861, 337734048, 3105588629, 29877483743, 299906019892, 3133423928557, 34002824654365, 382507638525838, 4452923233600903, 53561431659306039, 664728428775177890, 8500763141347126563, 111886109022440334593, 1513989730079050155936

Offset: 0

Ilya Gutkovskiy, Jan 21 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..553

Cf. A000110, A000129, A263575, A263576, A264037.

b:= proc(n, m) option remember; `if`(n=0,
      (<<2|1>, <1|0>>^m)[1, 2], m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jun 23 2023

FullSimplify[nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] Sinh[Sqrt[2] (Exp[x] - 1)]/Sqrt[2], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS2[n, k] Fibonacci[k, 2], {k, 0, n}], {n, 0, 24}]
Table[Sum[Binomial[n, k] BellB[n - k] (BellB[k, Sqrt[2]] - BellB[k, -Sqrt[2]])/(2 Sqrt[2]), {k, 0, n}], {n, 0, 24}]

E.g.f.: exp(exp(x) - 1)*sinh(sqrt(2)*(exp(x) - 1))/sqrt(2).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A000129(k).
a(n) = Sum_{k=0..n} binomial(n,k)*A000110(n-k)*A264037(k).

A323632 Stirling transform of Jacobsthal numbers (A001045).

Original entry on oeis.org

0, 1, 2, 7, 31, 152, 813, 4741, 29956, 203305, 1470795, 11276718, 91221419, 775677177, 6910797962, 64326920851, 623981351195, 6293426736344, 65867162316433, 714062197266081, 8005397253530924, 92676194887133693, 1106385117766336919, 13603803900252612966, 172082332173918135687

Offset: 0

Ilya Gutkovskiy, Jan 21 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..558

Cf. A000587, A001045, A001861, A263575, A263576.

b:= proc(n, m) option remember;
     `if`(n=0, round(2^m/3), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Aug 06 2021

nmax = 24; CoefficientList[Series[(Exp[2 (Exp[x] - 1)] - Exp[1 - Exp[x]])/3, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] (2^k - (-1)^k)/3, {k, 0, n}], {n, 0, 24}]
Table[(BellB[n, 2] - BellB[n, -1])/3, {n, 0, 24}]

E.g.f.: (exp(2*(exp(x) - 1)) - exp(1 - exp(x)))/3.
a(n) = Sum_{k=0..n} Stirling2(n,k)*A001045(k).
a(n) = (A001861(n) - A000587(n))/3.

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A263575 Stirling transform of Lucas numbers (A000032).

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A320352 Expansion of e.g.f. (exp(x) - 1)/(exp(x) - exp(2*x) + 1).

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A323620 Expansion of e.g.f. 2*sqrt(1 + x)*sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).

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A323631 Stirling transform of Pell numbers (A000129).

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A323632 Stirling transform of Jacobsthal numbers (A001045).

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A323620 Expansion of e.g.f. 2sqrt(1 + x)sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).