A006043 -id:A006043 - cp's OEIS frontend

A027472 Third convolution of the powers of 3 (A000244).

1, 9, 54, 270, 1215, 5103, 20412, 78732, 295245, 1082565, 3897234, 13817466, 48361131, 167403915, 573956280, 1951451352, 6586148313, 22082967873, 73609892910, 244074908070, 805447196631, 2646469360359, 8661172452084, 28242953648100, 91789599356325, 297398301914493, 960825283108362, 3095992578904722

Offset: 3

Olivier Gérard

Third column of A027465.

With offset = 2, a(n) is the number of length n words on alphabet {u,v,w,z} such that each word contains exactly 2 u's. - Zerinvary Lajos, Dec 29 2007

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A000244, A006043, A027465, A075513, A152818.

Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), A001788 (q=2), this sequence (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

[3^(n-3)*Binomial(n-1, 2): n in [3..40]]; // G. C. Greubel, May 12 2021

nn=41; Drop[Range[0,nn]!CoefficientList[Series[Exp[x]^3 x^2/2!,{x,0,nn}],x],2] (* Geoffrey Critzer, Oct 03 2013 *)
LinearRecurrence[{9,-27,27}, {1,9,54}, 40] (* G. C. Greubel, May 12 2021 *)
Abs[Take[CoefficientList[Series[1/(1+3x^2)^3,{x,0,60}],x],{1,-1,2}]] (* Harvey P. Dale, Mar 03 2022 *)

a(n)=([0,1,0; 0,0,1; 27,-27,9]^(n-3)*[1;9;54])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[3^(n-3)*binomial(n-1,2) for n in range(3, 40)] # Zerinvary Lajos, Mar 10 2009

Numerators of sequence a[3,n] in (b^2)[i,j]) where b[i,j] = binomial(i-1, j-1)/2^(i-1) if j <= i, 0 if j > i.
From Wolfdieter Lang: (Start)
a(n) = 3^(n-3)*binomial(n-1, 2).
G.f.: (x/(1-3*x))^3. (Third convolution of A000244, powers of 3.) (End)
a(n) = |A075513(n, 2)|/9, n >= 3.
a(n) = A152818(n-3,2)/2 = A006043(n-3)/2. - Paul Curtz, Jan 07 2009
The sequence 0, 1, 9, 54, ... has e.g.f.: (x + 3*x^2/2)*exp(3*x)/. - Paul Barry, Jul 23 2003
E.g.f.: E(0) where E(k) = 1 + 3*(2*k+3)*x/((2*k+1)^2 - 3*x*(k+2)*(2*k+1)^2/(3*x*(k+2) + 2*(k+1)^2/E(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 23 2012
With offset=2 e.g.f.: x^2*exp(3*x)/2. - Geoffrey Critzer, Oct 03 2013
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=3} 1/a(n) = 6 - 12*log(3/2).
Sum_{n>=3} (-1)^(n+1)/a(n) = 24*log(4/3) - 6. (End)

Corrected by T. D. Noe, Nov 07 2006
Better name from Wolfdieter Lang
Terms a(23) onward added by G. C. Greubel, May 12 2021

A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320

Offset: 0

Paul Curtz, Dec 13 2008

A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.

Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. - Omar E. Pol, Jan 06 2009

			From _Omar E. Pol_, Jan 06 2009: (Start)
Array begins:
  1,    1,      2,        6,         24,          120, ...
  1,    4,     18,       96,        600,         4320, ...
  1,   12,    108,      960,       9000,        90720, ...
  1,   32,    540,     7680,     105000,      1451520, ...
  1,   80,   2430,    53760,    1050000,     19595520, ...
  1,  192,  10206,   344064,    9450000,    235146240, ...
  1,  448,  40824,  2064384,   78750000,   2586608640, ...
  1, 1024, 157464, 11796480,  618750000,  26605117440, ...
  1, 2304, 590490, 64880640, 4640625000, 259399895040, ... (End)
Antidiagonal triangle:
  1;
  1,   1;
  1,   4,     2;
  1,  12,    18,     6;
  1,  32,   108,    96,     24;
  1,  80,   540,   960,    600,   120;
  1, 192,  2430,  7680,   9000,  4320,   720;
  1, 448, 10206, 53760, 105000, 90720, 35280, 5040;

G. C. Greubel, Antidiagonals n = 0..50, flattened
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. See page 422.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.

Cf. A000012, A000142, A001563, A001787, A006043, A006044, A009998.
Cf. A055533, A072597 (antidiagonal sums), A089148, A119502, A152650.
Cf. A152656, A154372.

A152818:= func< n,k | (k+1)^(n-k)*Factorial(k)*Binomial(n,k) >;
[A152818(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2023

len= 45; m= 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 +#2[[1]]^2 +(#2[[2]] -1)*#2[[2]] +#2[[1]]*(2*#2[[2]] -3))/ 2, #1}&, Table[(k+1)^n*(n+k)!/n!, {n,0,m}, {k,0,m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-François Alcover, May 27 2011 *)
T[n_, k_]:= (k+1)^(n-k)*k!*Binomial[n, k];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2023 *)

A(n,k) = (k+1)^n*(n+k)!/n! \\ Charles R Greathouse IV, Sep 10 2016

def A152818_row(n):
    R. = ZZ[]
    P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
    return P.coefficients()
for n in (0..12): print(A152818_row(n))  # Peter Luschny, May 03 2013

E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2! + (1+12*t+18*t^2+6*t^3)*x^3/3! + .... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011
From Peter Bala, Oct 09 2011: (Start)
From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650.
Sum_{k=0..n} T(n, k) = A072597(n) (antidiagonal sums). (End)
From G. C. Greubel, Apr 10 2023: (Start)
T(n, k) = (k+1)^(n-k) * k! * binomial(n, k) (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A089148(n). (End)

Better definition, extended and edited by Omar E. Pol and N. J. A. Sloane, Jan 05 2009

A006044 a(n) = 4^(n-4)(n-1)(n-2)*(n-3).

Original entry on oeis.org

6, 96, 960, 7680, 53760, 344064, 2064384, 11796480, 64880640, 346030080, 1799356416, 9160359936, 45801799680, 225485783040, 1095216660480, 5257039970304, 24970939858944, 117510305218560, 548381424353280, 2539871860162560, 11683410556747776, 53409876830846976

Offset: 4

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
Frank A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
Frank A. Haight, Letter to N. J. A. Sloane, n.d..
Index entries for linear recurrences with constant coefficients, signature (16,-96,256,-256).

Cf. A000142, A006043, A038846, A154120.

Column k=3 of square array A152818. - Paul Curtz, Dec 17 2008 [corrected by Omar E. Pol, Jan 07 2009]

[4^(n-4)*(n-3)*(n-2)*(n-1): n in [4..30]]; // Vincenzo Librandi, Aug 14 2011

a[n_] := 4^(n - 4)*(n - 1)*(n - 2)*(n - 3); Array[a, 25, 4] (* Amiram Eldar, Jan 08 2023 *)

G.f. = 6*x^4/(1-4*x)^4. - Emeric Deutsch, Apr 29 2004
a(n) = 6*A038846(n). - R. J. Mathar , Mar 22 2013
E.g.f.: (3 + exp(4*x)*(32*x^3 - 24*x^2 + 12*x - 3))/128. - Stefano Spezia, Jan 01 2023
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=4} 1/a(n) = 18*log(4/3) - 5.
Sum_{n>=4} (-1)^n/a(n) = 50*log(5/4) - 11. (End)

More terms from Emeric Deutsch, Apr 29 2004
Erroneous reference deleted by Martin J. Erickson (erickson(AT)truman.edu), Nov 03 2010
Entry revised by N. J. A. Sloane, Dec 27 2021

A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 9, 1, 1, 32, 54, 16, 1, 1, 80, 270, 160, 25, 1, 1, 192, 1215, 1280, 375, 36, 1, 1, 448, 5103, 8960, 4375, 756, 49, 1, 1, 1024, 20412, 57344, 43750, 12096, 1372, 64, 1

Offset: 0

Paul Curtz, Jan 08 2009

From A152650/A152656,coefficients of other exponential polynomials(*). a(n) is triangle A152818 where terms of each column is divided by the beginning one. See A000004, A001787(n+1), A006043=2*A027472, A006044=6*A038846.
(*) Not factorial as written in A006044. See A000110, Bell-Touchard. Second diagonal is 1,4,9,16,25, denominators of Lyman's spectrum of hydrogen, A000290(n+1) which has homogeneous indices for denominators series of Rydberg-Ritz spectrum of hydrogen.
The matrix inverse starts
      1;
     -1,      1;
      3,     -4,     1;
    -16,     24,    -9,     1;
    125,   -200,    90,   -16,   1;
  -1296,   2160, -1080,   240, -25,   1;
  16807, -28812, 15435, -3920, 525, -36, 1;
.. compare with A122525 (row reversed). - R. J. Mathar, Mar 22 2013
From Peter Bala, Jan 14 2015: (Start)
Exponential Riordan array [exp(z), z*exp(z)]. This triangle is the particular case a = 0, b = 1, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. Cf. A059297.
This is the triangle of connection constants when expressing the monomials x^n as a linear combination of the basis polynomials (x - 1)*(x - k - 1)^(k-1), k = 0,1,2,.... For example, from row 3 we have x^3 = 1 + 12*(x - 1) + 9*(x - 1)*(x - 3) + (x - 1)*(x - 4)^2.
Let M be the infinite lower unit triangular array with (n,k)-th entry (k*(n - k + 1) + 1)/(k + 1)*binomial(n,k). M is the row reverse of A145033. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle. See the Example section. (End)
T(n,k) is also the number of idempotent partial transformations of {1,2,...,n} having exactly k fixed points. - Geoffrey Critzer, Nov 25 2021

			With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/1      \ /1        \ /1        \      /1        \
|1 1     ||0 1       ||0 1      |      |1  1      |
|1 3 1   ||0 1 1     ||0 0 1    |... = |1  4  1   |
|1 6 5 1 ||0 1 3 1   ||0 0 1 1  |      |1 12  9  1|
|...     ||0 1 6 5 1 ||0 0 1 3 1|      |...       |
|...     ||...       ||...      |      |          |
- _Peter Bala_, Jan 13 2015

G. C. Greubel, Table of n, a(n) for n = 0..1274
Peter Bala, A 3 parameter family of generalized Stirling numbers
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

Cf. A080108, A152818, A059297, A145033.

/* As triangle */ [[(k+1)^(n-k)*Binomial(n,k) : k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 15 2016

T[n_, k_] := (k + 1)^(n - k)*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Sep 15 2016 *)

T(n,k) = (k+1)^(n-k)*binomial(n,k). k!*T(n,k) gives the entries for A152818 read as a triangular array.
E.g.f.: exp(x*(1+t*exp(x))) = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + (1+12*t+9*t^2+t*3)*x^3/3! + .... O.g.f.: Sum_{k>=1} (t*x)^(k-1)/(1-k*x)^k = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... Row sums are A080108. - Peter Bala, Oct 09 2011
From Peter Bala, Jan 14 2015: (Start)
Recurrence equation: T(n+1,k+1) = T(n,k+1) + Sum_{j = 0..n-k} (j + 1)*binomial(n,j)*T(n-j,k) with T(n,0) = 1 for all n.
Equals the matrix product A007318 * A059297. (End)

A154128 a(n) = 5^n*(n+4)!/n!.

Original entry on oeis.org

24, 600, 9000, 105000, 1050000, 9450000, 78750000, 618750000, 4640625000, 33515625000, 234609375000, 1599609375000, 10664062500000, 69726562500000, 448242187500000, 2838867187500000, 17742919921875000, 109588623046875000

Offset: 0

Omar E. Pol, Jan 05 2009

Column 4 of square array A152818.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (25, -250, 1250, -3125, 3125).

Cf. A000142, A006043, A006044, A152818, A154120.

[5^n*(n+4)*(n+3)*(n+2)*(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

LinearRecurrence[{25, -250, 1250, -3125, 3125}, {24, 600, 9000, 105000, 1050000}, 25] (* or *) Table[5^n*(n+4)*(n+3)*(n+2)*(n+1), {n,0,25}] (* G. C. Greubel, Sep 02 2016 *)

a(n) = 5^n*(n+4)*(n+3)*(n+2)*(n+1).
From R. J. Mathar, Feb 06 2009: (Start)
a(n) = A052762(n+4)*A000351(n).
a(n) = 24*A036071(n).
G.f: 24/(1-5*x)^5. (End)
From G. C. Greubel, Sep 02 2016: (Start)
a(n) = 25*a(n-1) - 250*a(n-2) + 1250*a(n-3) - 3125*a(n-4) + 3125*a(n-5).
E.g.f.: (24 + 480*x + 1800*x^2 + 2000*x^3 + 625*x^4)*exp(5*x). (End)

More terms from R. J. Mathar, Feb 06 2009

cp's OEIS Frontend

A027472 Third convolution of the powers of 3 (A000244).

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A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

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A006044 a(n) = 4^(n-4)*(n-1)*(n-2)*(n-3).

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A154372 Triangle T(n,k) = (k+1)^(n-k)*binomial(n,k).

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A154128 a(n) = 5^n*(n+4)!/n!.

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Magma

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Extensions

A006044 a(n) = 4^(n-4)(n-1)(n-2)*(n-3).