A053486 -id:A053486 - cp's OEIS frontend

A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 9, 1, 4, 10, 16, 24, 44, 1, 5, 17, 38, 65, 120, 265, 1, 6, 26, 78, 168, 326, 720, 1854, 1, 7, 37, 142, 393, 872, 1957, 5040, 14833, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 133496, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1334961

Offset: 0

Philippe Deléham, Dec 12 2003

Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018

			n\k -1   0   1    2    3    4     5     6  ...
----------------------------------------------
0  | 1,  1,  1,   1,   1,   1,    1,    1, ...
1  | 0,  1,  2,   3,   4,   5,    6,    7, ...
2  | 1,  2,  5,  10,  17,  26,   37,   50, ...
3  | 2,  6, 16,  38,  78, 152,  236,  366, ...
4  | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
...

Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.
Main diagonal gives A217701.
Cf. A080955, A008290, A292977.

(* Assuming offset (0, 0): *)
T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, Dec 24 2021 *)

For n > 0, k >= -1, T(n,k) is the permanent of the n X n matrix with k+1 on the diagonal and 1 elsewhere.
T(0,k) = 1.
T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.
T(n,k) = n*T(n-1, k) + k^n .
T(n,k) = n! * Sum_{j=0..n} k^j/j!.
E.g.f. for k-th column: exp(k*x)/(1-x).
Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - Peter Luschny, Dec 24 2021

Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018

A263823 a(n) = n!*Sum_{k=0..n} Fibonacci(k-1)/k!, where Fibonacci(-1) = 1, Fibonacci(n) = A000045(n) for n>=0.

Original entry on oeis.org

1, 1, 3, 10, 42, 213, 1283, 8989, 71925, 647346, 6473494, 71208489, 854501957, 11108525585, 155519358423, 2332790376722, 37324646028162, 634518982479741, 11421341684636935, 217005492008104349, 4340109840162091161, 91142306643403921146, 2005130746154886276158

Offset: 0

Vladimir Reshetnikov, Oct 27 2015

nonn

			For n = 3, a(3) = 3!*(Fibonacci(-1)/0! + Fibonacci(0)/1! + Fibonacci(1)/2! + Fibonacci(2)/3!) = 6*(1 + 0 + 1/2 + 1/6) = 10.
For n = 5, Gamma(5+1, phi)*exp(phi) = 120*sqrt(5) + 333 = 240*phi + 213, so a(5) = 213.
G.f. = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 213*x^5 + 1283*x^6 + 8989*x^7 + 71925*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..445
Eric Weisstein's MathWorld, Incomplete Gamma Function.
Eric Weisstein's MathWorld, Fibonacci Number.
Eric Weisstein's MathWorld, Golden Ratio.

Cf. A000045, A111139.

Cf. A009102, A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).

Table[n! Sum[Fibonacci[k-1]/k!, {k, 0, n}], {n, 0, 22}]
Round@Table[(E^(1-GoldenRatio) GoldenRatio Gamma[n+1, 1-GoldenRatio] + E^GoldenRatio Gamma[n+1, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 22}]

a(n) = (Gamma(n+1, 1-phi)*exp(1-phi)*phi+Gamma(n+1, phi)*exp(phi)/phi)/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function, phi=(1+sqrt(5))/2.
a(n) = (phi^(n-1)*hypergeom([1,-n], [], 1-phi)-(-phi)^(1-n)*hypergeom([1,-n], [], phi))/sqrt(5).
Gamma(n+1, phi)*exp(phi) = A111139(n)*phi + a(n).
E.g.f.: (exp(phi*x)/phi+exp(-x/phi)*phi)/(sqrt(5)*(1-x)) = exp(x/2)*(cosh(x*sqrt(5)/2)-sinh(x*sqrt(5)/2)/sqrt(5))/(1-x).
Recurrence: a(0) = 1, a(1) = 1, a(2) = 3, a(n) = (n+1)*a(n-1)+(2-n)*a(n-2)+(2-n)*a(n-3).
a(n) ~ 2*exp(phi-n)*n^(n+1/2)*(1+exp(-sqrt(5))*phi^2)*sqrt(Pi/10)/phi.
0 = a(n)*(+a(n+1) + a(n+2) - 4*a(n+3) + a(n+4)) + a(n+1)*(+a(n+1) + 3*a(n+2) - 5*a(n+3) + a(n+4)) + a(n+2)*(+2*a(n+2) - a(n+4)) + a(n+3)*(+a(n+3)) if n>=0. - Michael Somos, Oct 30 2015

A327997 Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).

Original entry on oeis.org

1, 3, 1, 9, 7, 1, 27, 38, 12, 1, 81, 192, 101, 18, 1, 243, 969, 755, 215, 25, 1, 729, 5115, 5494, 2205, 400, 33, 1, 2187, 29322, 40971, 21469, 5355, 679, 42, 1, 6561, 187992, 323658, 209356, 66619, 11452, 1078, 52, 1, 19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1

Offset: 0

Peter Luschny, Oct 27 2019

KummerU(-n, 1-n-x, 1) are the Charlier polynomials with coefficients in A094816, the coefficients of KummerU(-n, 1-n-x, 2) are in |A137346|.

The exponential generating function of this family of sequences of polynomials is in its general form (1-t)^(-x)*exp(alpha*t) with a parameter alpha.

			The triangle starts:
      1;
      3,       1;
      9,       7,       1;
     27,      38,      12,       1;
     81,     192,     101,      18,      1;
    243,     969,     755,     215,     25,      1;
    729,    5115,    5494,    2205,    400,     33,     1;
   2187,   29322,   40971,   21469,   5355,    679,    42,    1;
   6561,  187992,  323658,  209356,  66619,  11452,  1078,   52,  1;
  19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1;

A094816 (z=1), |A137346| (z=2), this sequence (z=3).
Columns k=0..3 give A000244, A346395, A381052, A382701.
Row sums in A053486.

egf := exp(3*t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(print(n!*seq(coeff(p(n), x, k), k=0..n)), n=0..9);

p [n_] := HypergeometricU[-n, 1 - n - x, 3];
Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten

T(n, k) = sum(j=k, n, 3^(n-j)*binomial(n, j)*abs(stirling(j, k, 1))); \\ Seiichi Manyama, Apr 19 2025

T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(3*t)*(1-t)^(-x).
From Igor Victorovich Statsenko, Feb 14 2025: (Start)
T(m, n, k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), for m = -3.
The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=-3. (End)

A346395 Expansion of e.g.f. -log(1 - x) * exp(3*x).

Original entry on oeis.org

0, 1, 7, 38, 192, 969, 5115, 29322, 187992, 1370745, 11392839, 107043606, 1122823944, 12989320785, 164040593067, 2243143392138, 32994768719376, 519229765892241, 8701862242296807, 154700700117472422, 2907409255935736752, 57588370882960384377, 1198954118077558162875

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A002104, A053486, A346394, A346396.

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[3 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[3^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+2)*v[i]-3*(i-1)*v[i-1]+3^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} 3^k / ((n-k) * k!).
a(n) ~ exp(3) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+2) * a(n-1) - 3 * (n-1) * a(n-2) + 3^(n-1). - Seiichi Manyama, May 27 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.

A108869 E.g.f. : exp(6x)/(1-x).

Original entry on oeis.org

1, 7, 50, 366, 2760, 21576, 176112, 1512720, 13781376, 134110080, 1401566976, 15780033792, 191537187840, 2503044135936, 35120982067200, 527284915992576, 8439379765788672, 143486382677852160, 2582856448158007296

Offset: 0

Philippe Deléham, Jul 13 2005

Binomial transform of A080954.

a(n) is the permanent of the n X n matrix with 7's on the diagonal and 1's elsewhere.

Amiram Eldar, Table of n, a(n) for n = 0..448

Cf. A000142, A000522, A008290, A010842, A053486, A053487, A080954.

a:=n->n!*sum(6^k/k!,k=0..n): seq(a(n),n=0..20); # Emeric Deutsch, Jul 18 2005
restart:F(x):=exp(6*x)/(1-x): f[0]:=F(x): for n from 1 to 20 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..18); # Zerinvary Lajos, Apr 03 2009

a[n_] := n! * Sum[6^k/k!, {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Jun 30 2020 *)

a(n) = n!*Sum_{ k = 0..n } 6^k/k!.
a(n) = Sum_{ k = 0..n } A008290(n, k)*7^k.
a(n) Sum_{ k = 0..n } k!*C(n, k)*6^(n-k).

More terms from Emeric Deutsch, Jul 18 2005

A111139 a(n) = n!*Sum_{k=0..n} Fibonacci(k)/k!.

Original entry on oeis.org

0, 1, 3, 11, 47, 240, 1448, 10149, 81213, 730951, 7309565, 80405304, 964863792, 12543229529, 175605213783, 2634078207355, 42145251318667, 716469272418936, 12896446903543432, 245032491167329389, 4900649823346594545

Offset: 0

Vladeta Jovovic, Oct 17 2005

Eigensequence of a triangle with the Fibonacci series as the left border, the natural numbers (1, 2, 3, ...) as the right border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's MathWorld, Incomplete Gamma Function.
Eric Weisstein's MathWorld, Fibonacci Number.
Eric Weisstein's MathWorld, Golden Ratio.

Cf. A000045, A263823.

Cf. A009102, A009551, A000142, A000166, A000522, A000023, A053486, A010844 (incomplete Gamma function values at other points).

a:=n->sum(fibonacci (j)*n!/j!,j=0..n):seq(a(n),n=0..20); # Zerinvary Lajos, Mar 19 2007

f[n_] := n!*Sum[Fibonacci[k]/k!, {k, 0, n}]; Table[ f[n], {n, 0, 20}] (* or *)
Simplify[ Range[0, 20]!CoefficientList[ Series[2/Sqrt[5]*Exp[x/2]*Sinh[Sqrt[5]*x/2]/(1 - x), {x, 0, 20}], x]] (* Robert G. Wilson v, Oct 21 2005 *)
Module[{nn=20,fibs,fct},fct=Range[0,nn]!;fibs=Accumulate[ Fibonacci[ Range[ 0,nn]]/fct];Times@@@Thread[{fct,fibs}]] (* Harvey P. Dale, Feb 19 2014 *)
Round@Table[(E^GoldenRatio Gamma[n+1, GoldenRatio] - E^(1-GoldenRatio) Gamma[n+1, 1-GoldenRatio])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 27 2015 *)

vector(100, n, n--; n!*sum(k=0, n, fibonacci(k)/k!)) \\ Altug Alkan, Oct 28 2015

E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2)/(1-x).
Recurrence: a(n) = (n+1)*a(n-1) - (n-2)*a(n-2) - (n-2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 2*sqrt(e/5)*sinh(sqrt(5)/2)*n!. - Vaclav Kotesovec, Oct 18 2012
From Vladimir Reshetnikov, Oct 27 2015: (Start)
Let phi=(1+sqrt(5))/2.
a(n) = (phi^n*hypergeom([1,-n], [], 1-phi)-(1-phi)^n*hypergeom([1,-n], [], phi))/sqrt(5).
a(n) = (exp(phi)*Gamma(n+1, phi)-exp^(1-phi)*Gamma(n+1, 1-phi))/sqrt(5), where Gamma(a, x) is the upper incomplete Gamma function.
Gamma(n+1, phi)*exp(phi) = a(n)*phi + A263823(n).
a(n) ~ exp(phi-n)*n^(n+1/2)*sqrt(2*Pi/5)*(1-exp(-sqrt(5))).
(End)

A336998 a(n) = n! * Sum_{d|n} 3^(d - 1) / d!.

Original entry on oeis.org

1, 5, 15, 87, 201, 3123, 5769, 148347, 913761, 11541123, 39975849, 2616723387, 6227552241, 230557039443, 4151870901369, 76980002233707, 355687471142721, 27886053280896963, 121645100796252489, 10474674957482235867, 135117295282596928401, 2811664555920692775603

Offset: 1

Ilya Gutkovskiy, Aug 10 2020

nonn

G. C. Greubel, Table of n, a(n) for n = 1..445

Cf. A034730, A053486, A057625, A336997.

A336998:= func< n | Factorial(n)*(&+[3^(d-1)/Factorial(d): d in Divisors(n)]) >;
[A336998(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

Table[n! Sum[3^(d - 1)/d!, {d, Divisors[n]}], {n, 1, 22}]
nmax = 22; CoefficientList[Series[Sum[(Exp[3 x^k] - 1)/3, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = n! * sumdiv(n, d, 3^(d-1)/d!); \\ Michel Marcus, Aug 12 2020

def A336998(n): return factorial(n)*sum(3^(k-1)/factorial(k) for k in (1..n) if (k).divides(n))
[A336998(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

E.g.f.: Sum_{k>=1} (exp(3*x^k) - 1) / 3.

a(p) = p! + 3^(p - 1), where p is prime.

A343686 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 4, 33, 410, 6796, 140824, 3501782, 101589732, 3368237928, 125634319104, 5206805098752, 237370661584704, 11805144854303760, 636030155604374400, 36903603627294958416, 2294156656214759133024, 152126925169297299197184, 10718105879980375520103936, 799564645068022035991527552

Offset: 0

Ilya Gutkovskiy, Apr 26 2021

nonn

Cf. A007840, A052820, A053486, A343685, A343687.

a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(1 - 3 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (1 - 3*x + log(1 - x)).

a(n) ~ n! / ((3/c + 2 - c) * (1 - c/3)^n), where c = LambertW(3*exp(2)) = 2.2761339297716461777892556270138... - Vaclav Kotesovec, Apr 26 2021

A296661 a(n) = (exp(k)Gamma(1+n, k) - exp(-k)Gamma(1+n, -k))/k! for k = 3.

Original entry on oeis.org

0, 1, 2, 15, 60, 381, 2286, 16731, 133848, 1211193, 12111930, 133290279, 1599483348, 20793814965, 291113409510, 4366705925619, 69867294809904, 1187744054815089, 21379392986671602, 406208467134180927, 8124169342683618540

Offset: 0

Peter Luschny, Dec 18 2017

nonn

Cf. A010843, A053486.

A296661 := n -> (exp(3)*GAMMA(1+n,3) - exp(-3)*GAMMA(1+n,-3))/6:
seq(simplify(A296661(n)), n=0..20);

a(n) = (A053486(n) - A010843(n+1))/6. [corrected by Jason Yuen, Nov 23 2024]

D-finite with recurrence a(n) -n*a(n-1) -9*a(n-2) +9*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 29 2020

cp's OEIS Frontend

A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

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A263823 a(n) = n!*Sum_{k=0..n} Fibonacci(k-1)/k!, where Fibonacci(-1) = 1, Fibonacci(n) = A000045(n) for n>=0.

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A327997 Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).

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Maple

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A346395 Expansion of e.g.f. -log(1 - x) * exp(3*x).

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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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A108869 E.g.f. : exp(6x)/(1-x).

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Extensions

A111139 a(n) = n!*Sum_{k=0..n} Fibonacci(k)/k!.

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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.

A296661 a(n) = (exp(k)Gamma(1+n, k) - exp(-k)Gamma(1+n, -k))/k! for k = 3.