A075500 -id:A075500 - cp's OEIS frontend

A075911 Third column of triangle A075500.

1, 30, 625, 11250, 188125, 3018750, 47265625, 728906250, 11133203125, 168996093750, 2554931640625, 38523925781250, 579858642578125, 8717878417968750, 130968170166015625, 1966522521972656250, 29517837677001953125, 442967564392089843750, 6646513462066650390625

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..2} A075513(3,m)*exp(5*(m+1)*x)/2!.

Colin Barker, Table of n, a(n) for n = 0..849
Index entries for linear recurrences with constant coefficients, signature (30,-275,750).

Cf. A016164, A075912.

LinearRecurrence[{30,-275,750},{1,30,625},30] (* Harvey P. Dale, Oct 06 2017 *)

Vec(1/((1-5*x)*(1-10*x)*(1-15*x)) + O(x^30)) \\ Colin Barker, Dec 11 2015

a(n) = A075500(n+3, 3) = (5^n)*S2(n+3, 3) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = (5^n - 8*10^n + 9*15^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 5*k*x).
E.g.f.: (d^3/dx^3)(((exp(5*x)-1)/5)^3)/3! = (exp(5*x) - 8*exp(10*x) + 9*exp(15*x))/2!.
a(n) = 30*a(n-1) - 275*a(n-2) + 750*a(n-3) for n > 2. - Colin Barker, Dec 11 2015

A075912 Fourth column of triangle A075500.

Original entry on oeis.org

1, 50, 1625, 43750, 1063125, 24281250, 532890625, 11386718750, 238867578125, 4946347656250, 101481884765625, 2068161621093750, 41943091064453125, 847579699707031250, 17082562164306640625, 343617765808105468750, 6901873153839111328125

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is (Sum_{m=0..3} A075513(4,m)*exp(5*(m+1)*x))/3!.

Colin Barker, Table of n, a(n) for n = 0..767
Index entries for linear recurrences with constant coefficients, signature (50,-875,6250,-15000).

Cf. A075911, A075913.

Table[5^n*(-1 + 3*2^(3+n) + 2^(6+2*n) - 3^(4+n))/6, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)

Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)) + O(x^30)) \\ Colin Barker, Dec 11 2015

a(n) = A075500(n+4, 4) = (5^n)*S2(n+4, 4) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = (Sum_{m=0..3}A075513(4, m)*((m+1)*5)^n, m=0..3)/3!.
G.f.: 1/Product_{k=1..4}(1-5*k*x).
E.g.f.: (d^4/dx^4)((((exp(5*x)-1)/5)^4)/4!) = (-exp(5*x) + 24*exp(10*x) - 81*exp(15*x) + 64*exp(20*x))/3!.
a(n) = 50*a(n-1) - 875*a(n-2) + 6250*a(n-3) - 15000*a(n-4) for n>3. - Colin Barker, Dec 11 2015

A075913 Fifth column of triangle A075500.

Original entry on oeis.org

1, 75, 3500, 131250, 4344375, 132890625, 3855156250, 107765625000, 2933008203125, 78271552734375, 2058270703125000, 53524929199218750, 1380066321044921875, 35349237725830078125, 900813505310058593750, 22863955398559570312500, 578500758117828369140625

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..4}(A075513(5,m)*exp(5*(m+1)*x))/4!.

Colin Barker, Table of n, a(n) for n = 0..714
Index entries for linear recurrences with constant coefficients, signature (75,-2125,28125,-171250,375000).

Cf. A000351, A016164, A075911, A075912, A075914.

Table[5^n*(1 - 2^(n+6) + 2*3^(n+5) - 4^(n+5) + 5^(n+4))/24, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)

Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

a(n) = A075500(n+5, 5) = (5^n)*S2(n+5, 5) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4}(A075513(5, m)*((m+1)*5)^n)/4!.
G.f.: 1/Product_{k=1..5}(1-5*k*x).
E.g.f.: (d^5/dx^5)((((exp(5*x)-1)/5)^5)/5!) = (exp(5*x) - 64*exp(10*x) + 486*exp(15*x) - 1024*exp(20*x) + 625*exp(25*x))/4!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)). - Colin Barker, Dec 12 2015

A075914 Sixth column of triangle A075500.

Original entry on oeis.org

1, 105, 6650, 330750, 14266875, 560896875, 20682062500, 728227500000, 24779833203125, 821666548828125, 26708267167968750, 854772944238281250, 27023254648193359375, 846046877171630859375, 26282219820458984375000, 811330550012329101562500

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..5}(A075513(6,m)*exp(5*(m+1)*x))/5!.

Colin Barker, Table of n, a(n) for n = 0..675
Index entries for linear recurrences with constant coefficients, signature (105,-4375,91875,-1015000,5512500,-11250000).

Cf. A000351, A016164, A075911, A075912, A075913, A075915.

Table[5^(n-1) * (-1 + 5*2^(5+n) + 5*2^(11+2*n) - 10*3^(5+n) - 5^(6+n) + 6^(5+n))/24, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)

Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

a(n) = A075500(n+6, 6) = (5^n)*S2(n+6, 6) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5}(A075513(6, m)*((m+1)*5)^n)/5!.
G.f.: 1/Product_{k=1..6}(1-5*k*x).
E.g.f.: (d^6/dx^6)((((exp(5*x)-1)/5)^6)/6!) = (-exp(5*x) + 160*exp(10*x) - 2430*exp(15*x) + 10240*exp(20*x) - 15625*exp(25*x) + 7776*exp(30*x))/5!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)). - Colin Barker, Dec 12 2015

A075915 Seventh column of triangle A075500.

Original entry on oeis.org

1, 140, 11550, 735000, 39991875, 1960612500, 89303500000, 3853850000000, 159664583203125, 6409926960937500, 251055710800781250, 9641722822265625000, 364483553427490234375, 13602971247133789062500, 502386213470141601562500, 18394848021467285156250000

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..6}(A075513(7,m)exp(5*(m+1)*x))/6!.

Colin Barker, Table of n, a(n) for n = 0..646
Index entries for linear recurrences with constant coefficients, signature (140,-8050,245000,-4230625,41037500,-204187500,393750000).

Cf. A000351, A016164, A075911, A075912, A075913, A075914.

Table[5^(n-1) * (1 - 3*2^(7 + n) - 5*2^(14 + 2*n) + 5*3^(7 + n) + 3*5^(7 + n) - 6^(7 + n) + 7^(6 + n))/144, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)

Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

a(n) = A075500(n+7, 7) = (5^n)S2(n+7, 7) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6}(A075513(7, m)*(5*(m+1))^n)/6!.
G.f.: 1/Product_{k=1..7}(1-5k*x).
E.g.f.: (d^7/dx^7)((((exp(5x)-1)/5)^7)/7!) = (exp(5*x) - 384*exp(10*x) + 10935*exp(15*x) - 81920*exp(20*x) + 234375*exp(25*x) - 279936*exp(30*x) + 117649*exp(35*x))/6!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)). - Colin Barker, Dec 12 2015

A005011 Shifts one place left under 5th-order binomial transform.

Original entry on oeis.org

1, 1, 6, 41, 331, 3176, 35451, 447981, 6282416, 96546231, 1611270851, 28985293526, 558413253581, 11458179765541, 249255304141006, 5725640423174901, 138407987170952351, 3510263847256823056

Offset: 0

N. J. A. Sloane, Simon Plouffe

Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+5 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=5, otherwise F(k+1)=F(k); see examples in A004211, A004212, and A004213, and Fxtbook link. [Joerg Arndt, Apr 30 2011]

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 = 0..66
Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.5, pp. 366-368
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Adalbert Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups<, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A075500 (row sums).

A004211 (RGS where s(k)<=F(k)+2), A004212 (s(k)<=F(k)+3), A004213 (s(k)<=F(k)+4), A000110 (s(k)<=F(k)+1). - Joerg Arndt, Apr 30 2011

Table[5^n BellB[n, 1/5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

x='x+O('x^66);
egf=exp(intformal(exp(5*x))); /* =  1 + x + 3*x^2 + 41/6*x^3 + 331/24*x^4 + ... */
/* egf=exp(1/5*(exp(5*x)-1)) */ /* alternative computation */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */

a(n) = Sum_{m=0..n} 5^(n-m)*Stirling2(n, m), n >= 0.
E.g.f.: exp((exp(5*x)-1)/5).
O.g.f. A(x) satisfies A'(x)/A(x) = exp(5*x).
E.g.f.: exp(Integral_{t = 0..x} exp(5*t)). - Joerg Arndt, Apr 30 2011
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1-5*j*x). - Joerg Arndt, Apr 30 2011
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n) = e^(-1/5)*5^{n-1}*f_n(1/5). - Milan Janjic, May 30 2008
a(n) = upper left term in M^n, M = an infinite square production matrix in which a diagonal of (5,5,5,...) is appended to the right of Pascal's triangle:
  1, 5, 0, 0, 0, ...
  1, 1, 5, 0, 0, ...
  1, 2, 1, 5, 0, ...
  1, 3, 3, 1, 5, ...
  ... - Gary W. Adamson, Jul 29 2011
G.f.: T(0)/(1-x), where T(k) = 1 - 5*x^2*(k+1)/( 5*x^2*(k+1) - (1-x-5*x*k)*(1-6*x-5*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
G.f. A(x) satisfies: A(x) = 1 + x*A(x/(1 - 5*x))/(1 - 5*x). - Ilya Gutkovskiy, May 03 2019
a(n) ~ 5^n * n^n * exp(n/LambertW(5*n) - 1/5 - n) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^n). - Vaclav Kotesovec, Jul 15 2021
From Peter Bala, Jun 29 2024: (Start)
a(n) = exp(-1/5)*Sum_{n >= 0} (5*n)^k/(n!*5^n).
Touchard's congruence holds: for all primes p != 5, a(p+k) == (a(k) + a(k+1)) (mod p) for k = 0,1,2,.... In particular, a(p) == 2 (mod p) for all primes p != 5. See A004211. (End)

A075499 Stirling2 triangle with scaled diagonals (powers of 4).

Original entry on oeis.org

1, 4, 1, 16, 12, 1, 64, 112, 24, 1, 256, 960, 400, 40, 1, 1024, 7936, 5760, 1040, 60, 1, 4096, 64512, 77056, 22400, 2240, 84, 1, 16384, 520192, 989184, 435456, 67200, 4256, 112, 1, 65536, 4177920, 12390400, 7956480, 1779456, 169344, 7392, 144, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(4*z) - 1)*x/4) - 1
Also the inverse Bell transform of the quadruple factorial numbers 4^n*n! (A047053) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			[1]; [4,1]; [16,12,1]; ...; p(3,x) = x(16 + 12*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*     1
*     4      1
*    16     12      1
*    64    112     24      1
*   256    960    400     40     1
*  1024   7936   5760   1040    60    1
*  4096  64512  77056  22400  2240   84   1
* 16384 520192 989184 435456 67200 4256 112 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000302, A016152, A019677, A075907-A075910. Row sums are A004213.

Cf. A075498, A075500.

Table[(4^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

for(n=1, 11, for(m=1, n, print1(4^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

# uses[inverse_bell_transform from A265605]
# Adds a column 1,0,0,... at the left side of the triangle.
multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
inverse_bell_matrix(multifact_4_4, 9) # Peter Luschny, Dec 31 2015

a(n, m) = (4^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*4)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 4m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-4k*x), m >= 1.
E.g.f. for m-th column: (((exp(4x)-1)/4)^m)/m!, m >= 1.

A016164 Expansion of 1/((1-5x)(1-10*x)).

Original entry on oeis.org

1, 15, 175, 1875, 19375, 196875, 1984375, 19921875, 199609375, 1998046875, 19990234375, 199951171875, 1999755859375, 19998779296875, 199993896484375, 1999969482421875, 19999847412109375, 199999237060546875

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..994
Index entries for linear recurrences with constant coefficients, signature (15,-50).

Second column of triangle A075500.

Cf. A008277, A016161, A075911.

[n le 2 select 15^(n-1) else 15*Self(n-1) -50*Self(n-2): n in [1..31]]; // G. C. Greubel, Nov 09 2024

Table[5^n*(2^(n+1)-1), {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{15,-50},{1,15},20] (* Harvey P. Dale, Aug 08 2023 *)

Vec(1/((1-5*x)(1-10*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

A016164=BinaryRecurrenceSequence(15,-50,1,15)
[A016164(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

a(n) = (5^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = -5^n + 2*10^n.
G.f.: 1/((1-5*x)*(1-10*x)).
E.g.f.: (d^2/dx^2)((((exp(5*x)-1)/5)^2)/2!) = -exp(5*x) + 2*exp(10*x).
Sum_{k=1..n} 5^(k-1)*5^(n-k)*binomial(n, k). - Zerinvary Lajos, Sep 24 2006
a(0)=1, a(n) = 10*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011

A075501 Stirling2 triangle with scaled diagonals (powers of 6).

Original entry on oeis.org

1, 6, 1, 36, 18, 1, 216, 252, 36, 1, 1296, 3240, 900, 60, 1, 7776, 40176, 19440, 2340, 90, 1, 46656, 489888, 390096, 75600, 5040, 126, 1, 279936, 5925312, 7511616, 2204496, 226800, 9576, 168, 1, 1679616, 71383680

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.

The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(6*z) - 1)*x/6) - 1.

			[1]; [6,1]; [36,18,1]; ...; p(3,x) = x(36 + 18*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      6       1
*     36      18       1
*    216     252      36       1
*   1296    3240     900      60      1
*   7776   40176   19440    2340     90    1
*  46656  489888  390096   75600   5040  126   1
* 279936 5925312 7511616 2204496 226800 9576 168 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 1-7 are A000400, A016175, A075916-A075920. Row sums are A005012.

Cf. A075500, A075502.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 6^n, 9); # Peter Luschny, Jan 28 2016

Flatten[Table[6^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[6^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

for(n=1, 11, for(m=1, n, print1(6^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (6^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*6)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 6m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-6k*x), m >= 1.
E.g.f. for m-th column: (((exp(6x)-1)/6)^m)/m!, m >= 1.

cp's OEIS Frontend

A075911 Third column of triangle A075500.

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A075912 Fourth column of triangle A075500.

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A075913 Fifth column of triangle A075500.

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A075914 Sixth column of triangle A075500.

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A075915 Seventh column of triangle A075500.

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A005011 Shifts one place left under 5th-order binomial transform.

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A075499 Stirling2 triangle with scaled diagonals (powers of 4).

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A016164 Expansion of 1/((1-5x)(1-10*x)).