A027471 -id:A027471 - cp's OEIS frontend

A084614 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 3x^2)^n.

1, 1, 1, -3, 1, 2, -5, -6, 9, 1, 3, -6, -17, 18, 27, -27, 1, 4, -6, -32, 19, 96, -54, -108, 81, 1, 5, -5, -50, 5, 211, -15, -450, 135, 405, -243, 1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729, 1, 7, 0, -91, -91, 546, 637, -2015, -1911, 4914, 2457, -7371, 0, 5103, -2187, 1, 8, 4, -112, -182, 728, 1456

Offset: 0

Paul D. Hanna, Jun 01 2003

			Rows:
  1;
  1, 1, -3;
  1, 2, -5,  -6,   9;
  1, 3, -6, -17,  18,  27, -27;
  1, 4, -6, -32,  19,  96, -54,  -108,   81;
  1, 5, -5, -50,   5, 211, -15,  -450,  135,  405, -243;
  1, 6, -3, -70, -30, 366, 181, -1098, -270, 1890, -243, -1458, 729;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A002426, A027471, A084600 - A084613, A084615.

A084614:= func< n,k | (&+[Binomial(n, k-j)*Binomial(k-j, j)*(-3)^j: j in [0..k]]) >;
[A084614(n,k): k in [0..2*n], n in [0..15]]; // G. C. Greubel, Mar 25 2023

With[{eq= (1+x-3*x^2)}, Flatten[Table[CoefficientList[Expand[eq^n], x], {n,0,13}]]] (* G. C. Greubel, Mar 02 2017 *)

for(n=0,12, for(k=0,2*n,t=polcoeff((1+x-3*x^2)^n,k,x); print1(t",")); print(" "))

def A084614(n,k): return ( (1+x-3*x^2)^n ).series(x, 30).list()[k]
flatten([[A084614(n,k) for k in range(2*n+1)] for n in range(13)]) # G. C. Greubel, Mar 25 2023

From G. C. Greubel, Mar 25 2023: (Start)
T(n, k) = Sum_{j=0..k} binomial(n, k-j)*binomial(k-j, j)*(-3)^j, for 0 <= k <= 2*n.
T(n, 2*n) = (-3)^n.
T(n, 2*n-1) = (-1)^(n-1)*A027471(n+1), n >= 1.
Sum_{k=0..2*n} T(n, k) = (-1)^n.
Sum_{k=0..2*n} (-1)^k*T(n, k) = (-3)^n. (End)

A085708 Arithmetic derivative of 10^n.

Original entry on oeis.org

0, 7, 140, 2100, 28000, 350000, 4200000, 49000000, 560000000, 6300000000, 70000000000, 770000000000, 8400000000000, 91000000000000, 980000000000000, 10500000000000000, 112000000000000000, 1190000000000000000, 12600000000000000000, 133000000000000000000

Offset: 0

Reinhard Zumkeller, Jul 19 2003

a(n) = A003415(A011557(n)) = A008589(n)*A011557(n-1).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A001787, A003415, A008589, A011557, A027471.

[7*n*10^(n-1): n in [0..20]]; // Vincenzo Librandi, Jun 09 2011

Table[7*n*10^(n-1),{n,0,20}] (* Harvey P. Dale, Jul 09 2021 *)

a(n) = 7*n*10^(n-1).

G.f.: 7*x/(10*x-1)^2.

A133224 Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Remove (y,x) from B when (x,y) is in B and x <> y and let R35 denote the reduced set B. Then a(n) = the sum of the sizes of the union of x and y for every (x,y) in R35.

Original entry on oeis.org

0, 2, 14, 78, 400, 1960, 9312, 43232, 197120, 885888, 3934720, 17307136, 75509760, 327182336, 1409343488, 6039920640, 25770065920, 109522223104, 463857647616, 1958507577344, 8246342451200

Offset: 0

Ross La Haye, Dec 30 2007, Jan 03 2008

A082134 is the analogous sequence if "union" is replaced by "intersection" and A002697 is the analogous sequence if "union" is replaced by "symmetric difference". Here, X union Y = Y union X are considered as the same Cartesian product [Relation (37): U_Q(n) in document of Ross La Haye in reference], if we want to consider that X Union Y and Y Union X are two distinct Cartesian products, see A212698. [Bernard Schott, Jan 11 2013]

			a(2) = 14 because for P(A) = {{},{1},{2},{1,2}} |{} union {1}| = 1, |{} union {2}| = 1, |{} union {1,2}| = 2, |{1} union {2}| = 2, |{1} union {1,2}| = 2 and |{2} union {1,2}| = 2, |{} union {}| = 0, |{1} union {1}| = 1, |{2} union {2}| = 1, |{1,2} union {1,2}| = 2, which sums to 14.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Cf. A027471, A002697, A082134, A212698.

[n*(2^(n-2) + 3*2^(2*n-3)): n in [0..30]]; // Vincenzo Librandi, Jun 10 2011

LinearRecurrence[{12,-52,96,-64},{0,2,14,78},30] (* Harvey P. Dale, Jan 24 2019 *)

a(n) = n*(2^(n-2) + 3*2^(2*n-3)).
G.f.: 2*x*(7*x^2-5*x+1) / ((2*x-1)^2*(4*x-1)^2). [Colin Barker, Dec 10 2012]
E.g.f.: exp(2*x)*(1 + 3*exp(2*x))*x. - Stefano Spezia, Aug 04 2022

A217629 Triangle, read by rows, where T(n,k) = k!C(n, k)3^(n-k) for n>=0, k=0..n.

Original entry on oeis.org

1, 3, 1, 9, 6, 2, 27, 27, 18, 6, 81, 108, 108, 72, 24, 243, 405, 540, 540, 360, 120, 729, 1458, 2430, 3240, 3240, 2160, 720, 2187, 5103, 10206, 17010, 22680, 22680, 15120, 5040, 6561, 17496, 40824, 81648, 136080, 181440, 181440, 120960, 40320

Offset: 0

Vincenzo Librandi, Nov 10 2012

Triangle formed by the derivatives of x^n evaluated at x=3.
Sum(T(n,k), k=0..n) = A053486(n) (see the Formula section of A053486). Also:
first column:     A000244;
second column:    A027471;
third column:   2*A027472;
fourth column:  6*A036216;
fifth column:  24*A036217.

			Triangle begins:
1;
3,     1;
9,     6,     2;
27,    27,    18,     6;
81,    108,   108,    72,     24;
243,   405,   540,    540,    360,    120;
729,   1458,  2430,   3240,   3240,   2160,    720;
2187,  5103,  10206,  17010,  22680,  22680,   15120,   5040;
6561,  17496, 40824,  81648,  136080, 181440,  181440,  120960,  40320; etc.

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A000244, A027471, A027472, A036216, A036217, A053486, A090802, A218016, A218017.

[Factorial(n)/Factorial(n-k)*3^(n-k): k in [0..n], n in [0..10]];

Flatten[Table[n!/(n-k)!*3^(n-k), {n, 0, 10}, {k, 0, n}]]

T(n,k) = 3^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(3x)*x^k.

A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

Original entry on oeis.org

3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199

Offset: 1

Peter Luschny, Mar 03 2018

nonn

Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.

An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.

			Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1,         (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1),     (A299795) (x = 2, y = 2),
(3^p - 1)/2,     (A003462) (x = 1, y = 3),
3^p - 2^p,       (A135171) (x = 2, y = 3),
p*3^(p - 1),     (A027471) (x = 3, y = 3),
(4^p - 1)/3,     (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p,       (A005061) (x = 3, y = 4),
p*4^(p - 1),     (A002697) (x = 4, y = 4),
(p^p-1)/(p-1),   (A023037),
p^p,             (A000312, A051674).
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.

Peter Luschny, Table of n, a(n) for n = 1..10000
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Indices of the nonzero values of A300333.

Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.

using Primes
function isA300332(n)
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 2
    while k <= K
        if k == 7
            K = Int(floor(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
            n == r && return true
        end
        k = nextprime(k+1)
    end
    return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))

A318773 Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

Original entry on oeis.org

1, 3, 9, 27, 81, 1, 243, 6, 729, 27, 2187, 108, 6561, 405, 1, 19683, 1458, 9, 59049, 5103, 54, 177147, 17496, 270, 531441, 59049, 1215, 1, 1594323, 196830, 5103, 12, 4782969, 649539, 20412, 90, 14348907, 2125764, 78732, 540, 43046721, 6908733, 295245, 2835, 1, 129140163, 22320522, 1082565, 13608, 15

Offset: 0

Zagros Lalo, Sep 04 2018

The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in center-justified triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)
The coefficients in the expansion of 1/(1-3*x-x^4) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.035744112294..., when n approaches infinity.

			Triangle begins:
          1;
          3;
          9;
         27;
         81,        1;
        243,        6;
        729,       27;
       2187,      108;
       6561,      405,       1;
      19683,     1458,       9;
      59049,     5103,      54;
     177147,    17496,     270;
     531441,    59049,    1215,     1;
    1594323,   196830,    5103,    12;
    4782969,   649539,   20412,    90;
   14348907,  2125764,   78732,   540;
   43046721,  6908733,  295245,  2835,   1;
  129140163, 22320522, 1082565, 13608,  15;
  387420489, 71744535, 3897234, 61236, 135;
  ...

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

G. C. Greubel, Rows n = 0..120 of the irregular triangle, flattened
Zagros Lalo, Third layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3x)^n
Zagros Lalo, Third layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n

Row sums give A052917.
Cf. A013610, A027465.
Cf. A304236, A304249
Cf. A317496, A317497.
Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3).
Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), A304249 (q=2), A317497 (q=3), this sequence (q=4).

[3^(n-4*k)*Binomial(n-3*k,k): k in [0..Floor(n/4)], n in [0..24]]; // G. C. Greubel, May 12 2021

T[n_, k_]:= T[n, k] = 3^(n-4k)*(n-3k)!/((n-4k)! k!); Table[T[n, k], {n, 0, 17}, {k, 0, Floor[n/4]} ]//Flatten
T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, 3T[n-1, k] + T[n-4, k-1]]; Table[T[n, k], {n, 0, 17}, {k, 0, Floor[n/4]}]//Flatten

flatten([[3^(n-4*k)*binomial(n-3*k,k) for k in (0..n//4)] for n in (0..24)]) # G. C. Greubel, May 12 2021

T(n,k) = 3^(n-4*k) * (n-3*k)!/(k! * (n-4*k)!) where n >= 0 and 0 <= k <= floor(n/4).

A383136 a(n) = Sum_{k=0..n} k^2 * 2^(n-k) * binomial(n,k).

Original entry on oeis.org

0, 1, 8, 45, 216, 945, 3888, 15309, 58320, 216513, 787320, 2814669, 9920232, 34543665, 119042784, 406552365, 1377495072, 4634696961, 15496819560, 51526925037, 170465015160, 561372288561, 1841022163728, 6014703091725, 19581781196016, 63546645708225, 205608702558168

Offset: 0

Seiichi Manyama, Apr 17 2025

Cf. A027471, A383137, A383138, A383139.

PARI
```
a(n) = 3^(n-2)*n*(2+n);
```

a(n) = 3^(n-2) * n * (2 + n).

A383137 a(n) = Sum_{k=0..n} k^3 * 2^(n-k) * binomial(n,k).

Original entry on oeis.org

0, 1, 12, 87, 504, 2565, 11988, 52731, 221616, 898857, 3542940, 13640319, 51490728, 191141613, 699376356, 2527001955, 9030245472, 31955015889, 112093661484, 390132432423, 1348223301720, 4629287423061, 15802106905332, 53651151578187, 181257000301584

Offset: 0

Seiichi Manyama, Apr 17 2025

Cf. A027471, A383136, A383138, A383139.

PARI
```
a(n) = 3^(n-3)*n*(2+6*n+n^2);
```

a(n) = 3^(n-3) * n * (2 + 6*n + n^2).

A383138 a(n) = Sum_{k=0..n} k^4 * 2^(n-k) * binomial(n,k).

Original entry on oeis.org

0, 1, 20, 189, 1320, 7785, 41148, 201285, 929232, 4100625, 17452260, 72098829, 290521080, 1146082041, 4439303820, 16923738645, 63619864992, 236206924065, 867305334708, 3152957079645, 11359168737480, 40589657212041, 143957705302620, 507079568653029

Offset: 0

Seiichi Manyama, Apr 17 2025

Cf. A027471, A383136, A383137, A383139.

PARI
```
a(n) = 3^(n-4)*n*(-6+20*n+12*n^2+n^3);
```

a(n) = 3^(n-4) * n * (-6 + 20*n + 12*n^2 + n^3).

A383139 a(n) = Sum_{k=0..n} k^5 * 2^(n-k) * binomial(n,k).

Original entry on oeis.org

0, 1, 36, 447, 3768, 25725, 153468, 832923, 4213296, 20179449, 92510100, 409137399, 1755881064, 7345518453, 30059956332, 120676965075, 476358203232, 1852442299377, 7108046758404, 26948581794351, 101065091563800, 375297714478701, 1381124599327836, 5040775635099147

Offset: 0

Seiichi Manyama, Apr 17 2025

Cf. A027471, A383136, A383137, A383138.

a(n) = 3^(n-5)*n*(-30+10*n+80*n^2+20*n^3+n^4);

a(n) = 3^(n-5) * n * (-30 + 10*n + 80*n^2 + 20*n^3 + n^4).

cp's OEIS Frontend

A084614 Triangle, read by rows, where the n-th row lists the (2*n+1) coefficients of (1 + x - 3*x^2)^n.

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Magma

Mathematica

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SageMath

Formula

A085708 Arithmetic derivative of 10^n.

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Magma

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A133224 Let P(A) be the power set of an n-element set A and let B be the Cartesian product of P(A) with itself. Remove (y,x) from B when (x,y) is in B and x <> y and let R35 denote the reduced set B. Then a(n) = the sum of the sizes of the union of x and y for every (x,y) in R35.

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A217629 Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.

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Magma

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A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

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Julia

A318773 Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

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A383136 a(n) = Sum_{k=0..n} k^2 * 2^(n-k) * binomial(n,k).

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PARI

A084614 Triangle, read by rows, where the n-th row lists the (2n+1) coefficients of (1 + x - 3x^2)^n.

A217629 Triangle, read by rows, where T(n,k) = k!C(n, k)3^(n-k) for n>=0, k=0..n.