A000998 -id:A000998 - cp's OEIS frontend

A275934 Shifts 4 places left under binomial transform.

0, 0, 0, 1, 0, 0, 0, 1, 4, 10, 20, 36, 68, 166, 540, 1961, 7012, 23878, 78004, 250311, 815196, 2787806, 10232556, 40266382, 166608080, 708407020, 3046352440, 13161936881, 57188405288, 251328286460, 1125890398160, 5177570523461, 24539362719532, 119861818560962, 601299401594540, 3082695751138656, 16075855888601716, 85005009812011810, 455172001509369028, 2468935975119176601, 13584735197391443020

Offset: 0

Olivier Gérard, Aug 12 2016

Cf. A000998, A275935, A275936.

Sum_{i=0..n} binomial(n,i)*a(i) = a(n+4).

G.f. A(x) satisfies: A(x) = x^3 + x^4 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Jul 01 2021

A275935 Shifts 5 places left under binomial transform.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 5, 15, 35, 70, 127, 225, 455, 1260, 4555, 17760, 67265, 241015, 818705, 2666400, 8464210, 26791045, 87104270, 300213875, 1119214050, 4500888827, 19104042345, 83376236115, 366831787085, 1609394914730, 7015234913278, 30426949154855, 131992116224295, 577090099245575, 2565792536742865, 11698401074992087, 55012217948708040

Offset: 0

Olivier Gérard, Aug 12 2016

Cf. A000998, A275934.

Sum_{i=0..n} binomial(n,i)*a(i) = a(n+5).

G.f. A(x) satisfies: A(x) = x^4 + x^5 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Jul 01 2021

A275936 Shifts 6 places under binomial transform.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 810, 1464, 3262, 10269, 40404, 165635, 653580, 2439069, 8626470, 29121393, 94647798, 299273206, 933818700, 2935248294, 9557815564, 33225405312, 125646127446, 514785555355, 2232901148760, 9976014439674, 44944467146100, 201608952292578, 895062795448170

Offset: 0

Olivier Gérard, Aug 12 2016

Robert Israel, Table of n, a(n) for n = 0..800

Cf. A000998, A275934, A275935.

A:= Array(0..10000):
A[5]:= 1:
for n from 6 to 100 do
  A[n]:= add(binomial(n-6,i)*A[i],i=0..n-6);
od:
convert(A,list); # Robert Israel, Mar 04 2024

Sum_{i=0..n} binomial(n,i)*a(i) = a(n+6).

G.f. A(x) satisfies: A(x) = x^5 + x^6 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Jul 01 2021

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

Original entry on oeis.org

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).

A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2x)) / (1 - 2x).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 27, 83, 271, 971, 3865, 16879, 78985, 388385, 1987201, 10561385, 58443891, 337724057, 2040085491, 12862712499, 84357800063, 573182197539, 4021203303593, 29062345301487, 216129411635057, 1653180368063361, 13003920016983361, 105158133803473329

Offset: 0

Ilya Gutkovskiy, Feb 08 2022

nonn

Shifts 3 places left under 2nd-order binomial transform.

Cf. A000996, A000998, A004211, A007472, A210540, A351343, A351344, A351345.

nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Binomial[n - 3, k] 2^k a[n - k - 3], {k, 0, n - 3}]]; Table[a[n], {n, 0, 27}]

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

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}.

A351189 G.f. A(x) satisfies: A(x) = x^2 + x^3 * A(x/(1 + x)) / (1 + x).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, -3, 6, -9, 6, 27, -169, 645, -1995, 5122, -9570, 1242, 109739, -756648, 3733128, -15527337, 55626585, -161247102, 260402511, 1028417064, -14243992155, 102551438561, -595149283191, 3010031905815, -13336771020834, 48891499316016, -111677138548476

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

sign

Shifts 3 places left under inverse binomial transform.

Cf. A000998, A010741, A351188.

nmax = 31; A[] = 0; Do[A[x] = x^2 + x^3 A[x/(1 + x)]/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 0; a[2] = 1; 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) = a(1) = 0, a(2) = 1; a(n) = Sum_{k=0..n-3} (-1)^k * binomial(n-3,k) * a(n-k-3).

cp's OEIS Frontend

A275934 Shifts 4 places left under binomial transform.

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A275935 Shifts 5 places left under binomial transform.

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A275936 Shifts 6 places under binomial transform.

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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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Magma

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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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Magma

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A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2*x)) / (1 - 2*x).

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Mathematica

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

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A351189 G.f. A(x) satisfies: A(x) = x^2 + x^3 * A(x/(1 + x)) / (1 + x).

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Mathematica

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A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2x)) / (1 - 2x).