A027471 -id:A027471 - cp's OEIS frontend

A383140 Triangle read by rows: the coefficients of polynomials (1/3^(m-n)) * Sum_{k=0..m} k^n * 2^(m-k) * binomial(m,k) in the variable m.

1, 0, 1, 0, 2, 1, 0, 2, 6, 1, 0, -6, 20, 12, 1, 0, -30, 10, 80, 20, 1, 0, 42, -320, 270, 220, 30, 1, 0, 882, -1386, -770, 1470, 490, 42, 1, 0, 954, 7308, -15064, 2800, 5180, 952, 56, 1, 0, -39870, 101826, -39340, -61992, 29820, 14364, 1680, 72, 1, 0, -203958, -40680, 841770, -666820, -86940, 139440, 34020, 2760, 90, 1

Offset: 0

Seiichi Manyama, Apr 17 2025

			f_n(m) = (1/3^(m-n)) * Sum_{k=0..m} k^n * 2^(m-k) * binomial(m,k).
f_0(m) = 1.
f_1(m) =    m.
f_2(m) =  2*m +   m^2.
f_3(m) =  2*m + 6*m^2 + m^3.
Triangle begins:
  1;
  0,   1;
  0,   2,    1;
  0,   2,    6,   1;
  0,  -6,   20,  12,   1;
  0, -30,   10,  80,  20,  1;
  0,  42, -320, 270, 220, 30, 1;
  ...

Eric Weisstein's World of Mathematics, Falling Factorial

Columns k=0..1 give A000007, A179929(n-1).
Row sums give A133494.
Alternating row sums give A212846.
Cf. A027471, A383136, A383137, A383138, A383139.
Cf. A129062, A209849.

T(n, k) = sum(j=k, n, 3^(n-j)*stirling(n, j, 2)*stirling(j, k, 1));

def a_row(n):
    s = sum(3^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
    return expand(s).list()
for n in (0..10): print(a_row(n))

T(n,k) = Sum_{j=k..n} 3^(n-j) * Stirling2(n,j) * Stirling1(j,k).
T(n,k) = [x^k] Sum_{k=0..n} 3^(n-k) * Stirling2(n,k) * FallingFactorial(x,k).
E.g.f. of column k (with leading zeros): g(x)^k / k! with g(x) = log(1 + (exp(3*x) - 1)/3).

A067371 Arithmetic derivatives of 3-smooth numbers.

Original entry on oeis.org

0, 1, 1, 4, 5, 12, 6, 16, 32, 21, 44, 27, 80, 60, 112, 81, 192, 156, 108, 272, 216, 448, 384, 297, 640, 540, 405, 1024, 912, 756, 1472, 1296, 1053, 2304, 2112, 1836, 1458, 3328, 3024, 2592, 5120, 4800, 4320, 3645, 7424, 6912, 6156, 11264, 5103, 10752

Offset: 1

Reinhard Zumkeller, Mar 20 2002, revised: Jul 19 2003

nonn

			a(18) = A003415(A003586(18)) = A003415(72) = A003415(2^3*3^2) = (3*3+2*2)*2^(3-1)*3^(2-1) = (9+4)*2^2*3^1 = 13*4*3 = 156.
a(27) = A003415(A003586(27)) = A003415(243) = A003415(2^0*3^5) = (3*0+2*5)*2^(0-1)*3^(5-1) = ((0+10)/2)*3^4 = 5*81 = 405.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001787, A003415, A003586, A027471, A022328, A022329.

s = {}; m = 12; Do[n = 3^k; While[n <= 3^m, AppendTo[s, n]; n*=2], {k, 0, m}]; ad[1] = 0; ad[n_] := n * Total @ (Last[#]/First[#] & /@ FactorInteger[n]); ad /@ Union[s] (* Amiram Eldar, Jan 29 2020 *)

A003415(2^i+3^j) = (3*i + 2*j) * 2^(i-1) * 3^(j-1), i, j >=0.

a(n) = A003415(A003586(n)).

A169585 A000004 preceded by 1, 3.

Original entry on oeis.org

1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Klaus Brockhaus, Dec 02 2009

Inverse binomial transform of A016777; second inverse binomial transform of A053220; third inverse binomial transform of A027471 without first term; fourth inverse binomial transform of A081039.

Cf. A000004 (zero sequence), A016777 (3*n+1), A053220 ((3*n-1)*2^(n-2)), A027471 ((n-1)*3^(n-2)), A081039 ((3*n+4)*4^(n-1), a(0)=1, a(1)=7), A130706 (1, 2, 0, 0, 0, ...), A166926 (1, 2, 4, 0, 0, 0, ...), A130779 (1, 1, 2, 0, 0, 0, ...).

PARI
```
{concat([1, 3], vector(103))}
```

a(0) = 1, a(1) = 3, a(n) = 0 for n > 1.
G.f.: 1+3*x.
a(n) = 3^n mod 9. - Ridouane Oudra, Apr 09 2025

A081666 n*3^(n-1)+A081567(n).

Original entry on oeis.org

1, 4, 16, 62, 233, 855, 3083, 10978, 38746, 135924, 474955, 1655789, 5766389, 20080608, 69976772, 244166410, 853410637, 2988825507, 10490538559, 36905911166, 130139760590, 459970519296, 1629395348591, 5784362027257

Offset: 0

Paul Barry, Mar 26 2003

Binomial transform of A081663.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-44,75,-45).

Cf. A000045.

LinearRecurrence[{11,-44,75,-45},{1,4,16,62},30] (* Harvey P. Dale, Aug 06 2022 *)

a(n)=A027471(n-1)+A081567(n) G.f.: (1-7x+16x^2-13x^3)/((3x - 1)^2(5x^2-5x+1))

A094951 a(n) = A081038(n) + A077616(n).

Original entry on oeis.org

6, 31, 144, 621, 2538, 9963, 37908, 140697, 511758, 1830519, 6456024, 22497669, 77590386, 265189059, 899198172, 3027619377, 10130328342, 33705582543, 111577100832, 367662044061, 1206427402746, 3943553157531, 12845313733284

Offset: 1

Gary W. Adamson, May 26 2004

Performing the same operation but using the multiplier [1 0 0] yields [3^n 2*A027471(n+1) A077616(n)]. Example: M^4 * [1 0 0] = [81 216 324] where 324 = A077616(4) and 216/2 = 108 = A027471(5).

			a(3) = 144 = 81 + 63 = A081038(3) + A077616(3).
a(4) = 621 = 297 + 324 = A081038(4) + A077616(4).
a(4) = 621 since M^4 * [1 1 1] = [81 297 621] = [3^4 A081038(4), a(4)].

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A081038, A077616, A027471.

List([1..30], n-> 3^(n-2)*(9+7*n+2*n^2)); # G. C. Greubel, Jun 06 2019

[3^(n-2)*(9+7*n+2*n^2): n in [1..30]]; // G. C. Greubel, Jun 06 2019

a[n_] := (MatrixPower[{{3, 0, 0}, {2, 3, 0}, {1, 2, 3}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 23}] (* Robert G. Wilson v, Jun 05 2004 *)
Table[3^(n-2)*(9+7*n+2*n^2), {n,1,30}] (* G. C. Greubel, Jun 06 2019 *)

vector(30, n, 3^(n-2)*(9+7*n+2*n^2)) \\ G. C. Greubel, Jun 06 2019

[3^(n-2)*(9+7*n+2*n^2) for n in (1..30)] # G. C. Greubel, Jun 06 2019

a(n) = A081038(n) + A077616(n).
Let M = the 3 X 3 matrix [3 0 0 / 2 3 0 / 1 2 3]; then M^n * [1 1 1] = [3^n A081038(n) a(n)], where a(n) - A081038(n) = A077616(n).
From Colin Barker, Nov 09 2012: (Start)
a(n) = 3^(n-2)*(9 + 7*n + 2*n^2).
a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3).
G.f.: x*(6 - 23*x + 27*x^2)/(1-3*x)^3. (End)
E.g.f.: -1 + (1 + 3*x + 2*x^2)*exp(3*x). - G. C. Greubel, Jun 06 2019

Edited and extended by Robert G. Wilson v, Jun 05 2004

A118357 Triangle read by rows: T(n,k) is the number of ternary sequences of length n containing k subsequences 00 (n>=0, 0<=k<=max(0,n-1)).

Original entry on oeis.org

1, 3, 8, 1, 22, 4, 1, 60, 16, 4, 1, 164, 56, 18, 4, 1, 448, 188, 68, 20, 4, 1, 1224, 608, 248, 80, 22, 4, 1, 3344, 1920, 864, 312, 92, 24, 4, 1, 9136, 5952, 2928, 1152, 380, 104, 26, 4, 1, 24960, 18192, 9696, 4128, 1472, 452, 116, 28, 4, 1, 68192, 54976, 31536, 14400

Offset: 0

Emeric Deutsch, May 24 2006

Sum of entries in row n is 3^n (A000244). T(n,0) = A028859(n). T(n,1) = A073388(n-2). Sum(k*T(n,k),k=0..n-1) = (n-1)*3^(n-2) (A027471).

			T(4,2) = 4 because we have 0001, 0002, 1000 and 2000.
Triangle starts:
1;
3;
8,1;
22,4,1;
60,16,4,1;

Alois P. Heinz, n = 0..141, flattened

Cf. A000244, A028859, A073388, A027471.

G:=(1+(1-t)*z)/(1-(2+t)*z-2*(1-t)*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: 1; for n from 1 to 12 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form

nn=15;a=1/(1-2x);b=x/(1-y x)+1;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[a b/(1-2x^2/((1-y x)(1-2x))),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Nov 19 2012 *)

G.f.: G-1, where G = G(t,z) = [1+(1-t)z]/[1-(2+t)z-2(1-t)z^2]. G.f. of column k is z^(k+1)*(1-2z)^(k-1)/(1-2z-2z^2)^(k+1) (k>=1).

A120907 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having sum of the lengths of the drops equal to k (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21, their lengths being the differences 1, 2 and 1, respectively.

Original entry on oeis.org

1, 3, 6, 2, 1, 10, 10, 7, 15, 30, 31, 4, 1, 21, 70, 105, 36, 11, 28, 140, 294, 184, 76, 6, 1, 36, 252, 714, 696, 396, 78, 15, 45, 420, 1554, 2160, 1666, 566, 141, 8, 1, 55, 660, 3102, 5808, 5918, 2990, 995, 136, 19, 66, 990, 5775, 13992, 18348, 12746, 5615, 1280, 226

Offset: 0

Emeric Deutsch, Jul 15 2006

Row n has 2*floor(n/2)+1 terms (i.e. each of the rows 2n and 2n+1 has 2n+1 terms). Row sums are the powers of 3 (A000244). T(n,0)=A000217(n+1) (the triangular numbers). Sum(k*T(n,k),k>=0)=4(n-1)3^(n-2)=A120908(n)=4*A027471(n).

			T(4,3)=4 because we have 1020,2010,2021 and 2120.
Triangle starts:
1;
3;
6,2,1;
10,10,7;
15,30,31,4,1;
21,70,105,36,11;

Cf. A000244, A000217, A120908, A027471, A120906.

G:=1/(1-z+t*z)/(1-2*z+z^2-t*z-t*z^2): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..2*floor(n/2)) od; # yields sequence in triangular form

G.f.=G(t,z)=1/[(1-z+tz)(1-2z+z^2-tz-tz^2)].

A126186 Triangle read by rows: T(n,k) is number of hex trees with n edges and level of first leaf (in the preorder traversal) equal to k (1 <= k <= n).

Original entry on oeis.org

3, 1, 9, 3, 6, 27, 10, 19, 27, 81, 36, 66, 90, 108, 243, 137, 245, 325, 378, 405, 729, 543, 954, 1242, 1416, 1485, 1458, 2187, 2219, 3848, 4944, 5563, 5760, 5589, 5103, 6561, 9285, 15942, 20286, 22620, 23235, 22410, 20412, 17496, 19683, 39587, 67442, 85194

Offset: 1

Emeric Deutsch, Dec 22 2006

A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).

			Triangle starts:
   3;
   1,   9;
   3,   6,  27;
  10,  19,  27,  81;
  36,  66,  90, 108, 243;

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A000244, A002212, A025238, A027471, A126187.

G:=2/(2-t-3*t*z+t*sqrt(1-6*z+5*z^2))-1: Gser:=simplify(series(G,z=0,14)): for n from 1 to 10 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 10 do seq(coeff(P[n],t,k),k=1..n) od; # yields sequence in triangular form

n = 10; g[t_, z_] = 2/(2 - t - 3t*z + t*Sqrt[1 - 6z + 5z^2]) - 1; Flatten[ Rest[ CoefficientList[#, t]] & /@ Rest[ CoefficientList[ Series[g[t, z], {z, 0, n}], z]]] (* Jean-François Alcover, Jul 22 2011, after g.f. *)

T(n,k) = [k/(n-k)] sum(3^(2k+2j-n)*binomial(n-k,j)*binomial(k-1+j,n-k-1-j), j=ceiling((n-2k)/2)..n-k) if 1<=k

G.f.: 2/[2-t-3tz+t*sqrt(1-6z+5z^2)]-1.

Sum of terms in row n = A002212(n+1).

T(n,1) = A025238(n); T(n,1) = A002212(n-1) for n>=2.

T(n,n) = 3^n = A000244(n); T(n,n-1) = (n-1)*3^(n-2) = A027471(n) (n>=2).

Sum_{k=1..n} k*T(n,k) = A126187(n).

A181371 Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 3, 8, 1, 21, 6, 55, 25, 1, 144, 90, 9, 377, 300, 51, 1, 987, 954, 234, 12, 2584, 2939, 951, 86, 1, 6765, 8850, 3573, 480, 15, 17711, 26195, 12707, 2305, 130, 1, 46368, 76500, 43398, 10008, 855, 18, 121393, 221016, 143682, 40426, 4740, 183, 1, 317811

Offset: 0

Emeric Deutsch, Oct 31 2010

Row n contains 1 + floor(n/2) entries.
Sum of entries in row n is 3^n = A000244(n).
T(n,0) = F(2n+2) = A001906(n+1) (even-subscripted Fibonacci numbers).
T(n,1) = A001871(n-2).
Sum_{k>=0}k*T(n,k) = (n-1)*3^(n-2) = A027471(n) (n>=1).

			T(3,1)=6 because we have 010, 011, 012, 001, 101 and 201.
T(4,2)=1 because we have 0101.
Triangle starts:
    1;
    3;
    8,  1;
   21,  6;
   55, 25,  1;
  144, 90,  9;

Alois P. Heinz, Rows n = 0..200, flattened
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.

Cf. A000244, A001871, A001906, A027471.

G := 1/(1-3*z+z^2-t*z^2): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

G.f. = G(t,z) = 1/(1 - 3z + z^2 - tz^2).

A360984 Triangular array read by rows. T(n,k) is the number of idempotent Boolean relation matrices on [n] with exactly k reflexive points, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 4, 1, 27, 66, 29, 1, 108, 780, 1116, 355, 1, 405, 8020, 29250, 28405, 6942, 1, 1458, 76110, 649260, 1460425, 1068576, 209527

Offset: 0

Geoffrey Critzer, Feb 27 2023

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   6,    4;
  1,  27,   66,    29;
  1, 108,  780,  1116,   355;
  1, 405, 8020, 29250, 28405, 6942;
  ...

Cf. A121337 (row sums), A000798 (main diagonal).

Cf. A245767, A027471 (column 1).

T(n,n) = A245767(n,n) = A000798(n).
T(n,n-1) = A245767(n,n-1).
T(n,1) = n*Sum_k Sum_j binomial(n-1,k)*binomial(n-1-k,j) = A027471(n+1).
E.g.f. for column 1 is x*exp(x)^3.
E.g.f. for column 2 is x^2/2*exp(x)^3 + x^2*exp(x)^6 + x^2/2*exp(x)^7.
E.g.f. for column 3 is x^3/3!*exp(x)^15 + x^3/3!*exp(x)^3 + x^3*exp(x)^10 + x^3*exp(x)^12 + x^3/2!*exp(x)^7 + 2*x^3/2!*exp(x)^6 + 2*x^3/2*exp(x)^12.

Rows 5 and 6 added by Geoffrey Critzer, Mar 05 2023

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A383140 Triangle read by rows: the coefficients of polynomials (1/3^(m-n)) * Sum_{k=0..m} k^n * 2^(m-k) * binomial(m,k) in the variable m.

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A067371 Arithmetic derivatives of 3-smooth numbers.

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A169585 A000004 preceded by 1, 3.

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A081666 n*3^(n-1)+A081567(n).

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A094951 a(n) = A081038(n) + A077616(n).

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A118357 Triangle read by rows: T(n,k) is the number of ternary sequences of length n containing k subsequences 00 (n>=0, 0<=k<=max(0,n-1)).

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A120907 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having sum of the lengths of the drops equal to k (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21, their lengths being the differences 1, 2 and 1, respectively.

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A126186 Triangle read by rows: T(n,k) is number of hex trees with n edges and level of first leaf (in the preorder traversal) equal to k (1 <= k <= n).