A000997 -id:A000997 - cp's OEIS frontend

A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

0, 1, 1, 0, 1, 3, 6, 11, 23, 60, 179, 553, 1716, 5415, 17801, 61956, 228391, 882309, 3530322, 14531621, 61454091, 267479778, 1200680113, 5561767211, 26553471186, 130366882251, 656668581417, 3387887246292, 17886582294921, 96603394562849, 533645344137390, 3014295344076655

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..695

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346051, A346052.

nmax = 31; A[] = 0; Do[A[x] = x + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346050
    if (n<3): return (0,1,1)[n]
    else: return sum(binomial(n-3,k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 28 2022

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

A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 5, 12, 28, 68, 181, 531, 1671, 5491, 18627, 65299, 237880, 903907, 3580619, 14729777, 62639952, 274442521, 1236730244, 5729809348, 27292248240, 133614280479, 671803041553, 3464970976743, 18309428363425, 99010800275743, 547462187824465, 3093329527120022

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..695

Cf. A000994, A000995, A000996, A000997, A000998.

Cf. A007476, A210540, A346050, A346052.

function a(n)
  if n lt 3 then return (1+(-1)^n)/2;
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022

nmax = 31; A[] = 0; Do[A[x] = 1 + x^2 + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346051
    if (n<3): return (1, 0, 1)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

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

A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 5, 11, 29, 80, 222, 630, 1881, 6004, 20420, 72979, 270659, 1035590, 4087205, 16675630, 70440641, 307933393, 1390117953, 6462787357, 30871458702, 151298796000, 760250325004, 3915477534861, 20662363081756, 111662169790416, 617482470676567, 3490973387652861

Offset: 0

Ilya Gutkovskiy, Jul 02 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..650

Cf. A000994, A000995, A000996, A000997, A000998, A007476, A210540, A346050, A346051.

function a(n) // a = A346052
  if n lt 3 then return Floor((3-n)/2);
  else return (&+[Binomial(n-3,j)*a(j): j in [0..n-3]]);
  end if; return a;
end function;
[a(n): n in [0..35]]; // G. C. Greubel, Nov 30 2022

nmax = 31; A[] = 0; Do[A[x] = 1 + x + x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

@CachedFunction
def a(n): # a = A346052
    if (n<3): return (1, 1, 0)[n]
    else: return sum(binomial(n-3, k)*a(k) for k in range(n-2))
[a(n) for n in range(51)] # G. C. Greubel, Nov 30 2022

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

A088022 a(n) = floor(sum_{k>=0} k^n /(k!)^3); related to generalized Bell numbers.

Original entry on oeis.org

2, 1, 1, 2, 3, 6, 12, 28, 68, 176, 484, 1409, 4334, 14002, 47357, 167157, 614297, 2345730, 9290084, 38092233, 161436136, 706061825, 3182452003, 14764717643, 70429572474, 345075959701, 1734987079149, 8943648710357, 47228775626154

Offset: 0

Paul D. Hanna, Sep 19 2003

nonn

			a(8) = 68 = floor(17*2.1297 + 12*1.2641 + 11*1.5428) = floor(68.3463).

Cf. A086880, A000996, A000997, A000998.

B(n) := sum_{k>=0} k^n/(k!)^3 = A000996(n)*B(0) + A000997(n)*B(1) + A000998(n)*B(2) where B(0)=2.129702548983..., B(1)=1.264181150389..., B(2)=1.542838638501...; observe that these shift 3 places left under binomial transform: A000996={1, 0, 0, 1, 1, 1, 2, 6, 17, 44, 112, 304, 918, ...}, A000997={0, 1, 0, 0, 1, 2, 3, 5, 12, 36, 110, 326, 963, ...}, A000998={0, 0, 1, 0, 0, 1, 3, 6, 11, 24, 69, 227, 753, ...}; here A000998 is offset with 5 leading terms: {0, 0, 1, 0, 0}.

A351188 G.f. A(x) satisfies: A(x) = x + x^3 * A(x/(1 + x)) / (1 + x).

Original entry on oeis.org

0, 1, 0, 0, 1, -2, 3, -3, -2, 24, -94, 280, -687, 1270, -655, -9306, 65087, -306724, 1202250, -4033365, 10855578, -15470865, -69819687, 850568716, -5679272040, 30749200898, -144912453016, 593521998765, -1956641103972, 3349999455415, 20123433921282, -295760693980981

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 3 places left under inverse binomial transform.

Cf. A000997, A010741, A351189.

nmax = 31; A[] = 0; Do[A[x] = x + x^3 A[x/(1 + x)]/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[(-1)^k Binomial[n - 3, k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

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

cp's OEIS Frontend

A346050 G.f. A(x) satisfies: A(x) = x + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

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A346051 G.f. A(x) satisfies: A(x) = 1 + x^2 + x^3 * A(x/(1 - x)) / (1 - x).

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A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

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A088022 a(n) = floor(sum_{k>=0} k^n /(k!)^3); related to generalized Bell numbers.

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A351188 G.f. A(x) satisfies: A(x) = x + x^3 * A(x/(1 + x)) / (1 + x).

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