A000996 -id:A000996 - cp's OEIS frontend

A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

1, 2, 1, 5, 1, 1, 15, 2, 1, 1, 52, 5, 1, 1, 1, 203, 13, 2, 1, 1, 1, 877, 36, 6, 1, 1, 1, 1, 4140, 109, 17, 2, 1, 1, 1, 1, 21147, 359, 44, 7, 1, 1, 1, 1, 1, 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1, 678570, 4731, 304, 65, 8, 1, 1, 1, 1, 1, 1, 4213597, 18657, 918, 165, 30, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Alois P. Heinz, Sep 06 2008

The matrix inverse starts:
     1;
    -2,   1;
    -3,  -1,  1;
    -8,  -1, -1,  1;
   -31,  -3,  0, -1,  1;
  -132,  -7, -1,  0, -1,  1;
  -616, -19, -4,  0,  0, -1, 1; - R. J. Mathar, Mar 22 2013

			T(5,2) = 5, because [1,3,3,1] * [1,0,1,1] = 5.
Triangle begins:
:      1;
:      2,    1;
:      5,    1,   1;
:     15,    2,   1,  1;
:     52,    5,   1,  1, 1;
:    203,   13,   2,  1, 1, 1;
:    877,   36,   6,  1, 1, 1, 1;
:   4140,  109,  17,  2, 1, 1, 1, 1;
:  21147,  359,  44,  7, 1, 1, 1, 1, 1;
: 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1;

Alois P. Heinz, Rows n = 1..141, flattened
N. J. A. Sloane, Transforms

Columns 1-6 give: A000110, A000994, A000996, A010748, A010749, A010750.

Cf. A007318.

T:= proc(n, k) option remember; `if`(n

t[n_, k_] := t[n, k] = If[n < k, If[n == 0, 1, 0], Sum[Binomial[n-k, j]*t[j, k], {j, 0, n-k}]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 13}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

T(n,k) = Sum_{j=0..n-k} C(n-k,j)*T(j,k) if n>=k, else T(n,k) = 1 if n=1, else T(n,k) = 0.

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).

A336971 G.f. A(x) satisfies: A(x) = 1 - x^3 * A(x/(1 - x)) / (1 - x).

Original entry on oeis.org

1, 0, 0, -1, -1, -1, 0, 4, 15, 40, 86, 134, 16, -1060, -6119, -25187, -86678, -250846, -537819, -175233, 6998009, 55632942, 310923272, 1465146781, 6011047682, 20719304348, 49356093300, -36579100806, -1549214884054, -13807417413199, -92912464763743

Offset: 0

Ilya Gutkovskiy, Aug 09 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..704

Cf. A000587, A000996, A336970.

nmax = 30; A[] = 0; Do[A[x] = 1 - x^3 A[x/(1 - x)]/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 3, k] a[k], {k, 0, n - 3}]; Table[a[n], {n, 0, 30}]

a(0) = 1, a(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}.

cp's OEIS Frontend

A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down 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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A346052 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x/(1 - x)) / (1 - x).

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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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A336971 G.f. A(x) satisfies: A(x) = 1 - x^3 * A(x/(1 - x)) / (1 - 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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A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2x)) / (1 - 2x).