A075504 -id:A075504 - cp's OEIS frontend

A076013 Seventh column of triangle A075504.

1, 252, 37422, 4286520, 419818707, 37047106404, 3037410645984, 235940417032320, 17594974122819093, 1271468563282273356, 89638618747098243186, 6196581962116572990600, 421646012618644954061559

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..6} (A075513(7,m)*exp(9*(m+1)*x))/6!.

Michael De Vlieger, Table of n, a(n) for n = 0..554

Cf. A075504, A075513, A076012.

With[{m = 7}, Array[9^(# - m) StirlingS2[#, m] &, 13, m]] (* Michael De Vlieger, Dec 24 2017, after Indranil Ghosh at A075504 *)

a(n) = A075504(n+7, 7) = (9^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} (A075513(7, m)*((m+1)*9)^n)/6!.
G.f.: 1/Product_{k=1..7} (1 - 9*k*x).
E.g.f.: (d^7/dx^7)(((exp(9*x)-1)/9)^7)/7! = (exp(9*x) - 384*exp(18*x) + 10935*exp(27*x) - 81920*exp(36*x) + 234375*exp(45*x) - 279936*exp(54*x) + 117649*exp(63*x))/6!.

A076008 Second column of triangle A075504.

Original entry on oeis.org

1, 27, 567, 10935, 203391, 3720087, 67493007, 1219657095, 21996874431, 396331160247, 7137447668847, 128505439098855, 2313380333315871, 41643387865514007, 749603858371707087, 13493075341822822215

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..1} (A075513(3,m)*exp(9*(m+1)*x)).

Michael De Vlieger, Table of n, a(n) for n = 0..796
Index entries for linear recurrences with constant coefficients, signature (27,-162).

Cf. A001019, A076009.

CoefficientList[Series[1/((1-9x)(1-18x)),{x,0,30}],x] (* or *) LinearRecurrence[{27,-162},{1,27},30] (* Harvey P. Dale, Dec 01 2015 *)

a(n) = A075504(n+2, 2) = (9^n)*S2(n+2, 2) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = -9^n + 2*18^n.
G.f.: 1/((1-9*x)*(1-18*x)).
E.g.f.: (d^2/dx^2)(((exp(9*x)-1)/9)^2)/2! = -exp(9*x) + 2*exp(18*x).
a(0)=1, a(1)=27, a(n) = 27*a(n-1) - 162*a(n-2). - Harvey P. Dale, Dec 01 2015

A076009 Third column of triangle A075504.

Original entry on oeis.org

1, 54, 2025, 65610, 1974861, 57041334, 1607609025, 44625100770, 1226874595221, 33521945231214, 912229968911625, 24758714599712730, 670798674525559581, 18153207600055622694, 490886209059873519825

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..2} (A075513(3,m)*exp(9*(m+1)*x))/2!.

Michael De Vlieger, Table of n, a(n) for n = 0..698

Cf. A076008, A076010.

With[{m = 3}, Array[9^(# - m) StirlingS2[#, m] &, 15, m]] (* Michael De Vlieger, Dec 24 2017, after Indranil Ghosh at A075504 *)

a(n) = A075504(n+3, 3) = (9^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (9^n - 8*18^n + 9*27^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 9*k*x).
E.g.f.: (d^3/dx^3)(((exp(9*x)-1)/9)^3)/3! = (exp(9*x) - 8*exp(18*x) + 9*exp(27*x))/2!.

A076010 Fourth column of triangle A075504.

Original entry on oeis.org

1, 90, 5265, 255150, 11160261, 458810730, 18124795305, 697117731750, 26323112938221, 981154011007170, 36233774365169745, 1329174591745823550, 48521083977375207381, 1764912230785563088410, 64027726517340144702585

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..3} (A075513(4,m)*exp(9*(m+1)*x))/3!.

Michael De Vlieger, Table of n, a(n) for n = 0..641

Cf. A076009, A076011.

With[{m = 4}, Array[9^(# - m) StirlingS2[#, m] &, 15, m]] (* Michael De Vlieger, Dec 24 2017, after Indranil Ghosh at A075504 *)

a(n) = A075504(n+4, 4) = (9^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (-9^n + 24*18^n - 81*27^n + 64*36^n)/3!.
G.f.: 1/Product_{k=1..4} (1 - 9*k*x).
E.g.f.: (d^4/dx^4)(((exp(9*x)-1)/9)^4)/4! = (-exp(9*x) + 24*exp(18*x) - 81*exp(27*x) + 64*exp(36*x))/3!.

A076011 Fifth column of triangle A075504.

Original entry on oeis.org

1, 135, 11340, 765450, 45605511, 2511058725, 131122437930, 6597627438600, 323216347675221, 15525889656392115, 734898808902814920, 34399620992372494950, 1596504028634137480131, 73607593519321749694305

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..4} (A075513(5,m)*exp(9*(m+1)*x))/4!.

Michael De Vlieger, Table of n, a(n) for n = 0..604

Cf. A076010, A076012.

With[{m = 5}, Array[9^(# - m) StirlingS2[#, m] &, 14, m]] (* Michael De Vlieger, Dec 24 2017, after Indranil Ghosh at A075504 *)

a(n) = A075504(n+5, 5) = (9^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} (A075513(5, m)*(9*(m+1))^n)/4!.
G.f.: 1/Product_{k=1..5} (1 - 9*k*x).
E.g.f.: (d^5/dx^5)(((exp(9*x)-1)/9)^5)/5! = (exp(9*x) - 64*exp(18*x) + 486*exp(27*x) - 1024*exp(36*x) + 625*exp(45*x))/4!.

A076012 Sixth column of triangle A075504.

Original entry on oeis.org

1, 189, 21546, 1928934, 149767947, 10598527863, 703442942532, 44583546335328, 2730727849782933, 162985193544670497, 9536099260315021758, 549348981049383669882, 31261349005300855653759

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..5} (A075513(6,m)*exp(9*(m+1)*x))/5!.

Michael De Vlieger, Table of n, a(n) for n = 0..576

Cf. A076011, A076013.

With[{m = 6}, Array[9^(# - m) StirlingS2[#, m] &, 13, m]] (* Michael De Vlieger, Dec 24 2017, after Indranil Ghosh at A075504 *)

a(n) = A075504(n+6, 6) = (9^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} (A075513(6, m)*((m+1)*9)^n)/5!.
G.f.: 1/Product_{k=1..6} (1 - 9*k*x).
E.g.f.: (d^6/dx^6)(((exp(9*x)-1)/9)^6)/6! = (-exp(9*x) + 160*exp(18*x) - 2430*exp(27*x) + 10240*exp(36*x) - 15625*exp(45*x) + 7776*exp(54*x))/5!.

A075503 Stirling2 triangle with scaled diagonals (powers of 8).

Original entry on oeis.org

1, 8, 1, 64, 24, 1, 512, 448, 48, 1, 4096, 7680, 1600, 80, 1, 32768, 126976, 46080, 4160, 120, 1, 262144, 2064384, 1232896, 179200, 8960, 168, 1, 2097152, 33292288, 31653888, 6967296, 537600, 17024, 224, 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(8*z) - 1)*x/8) - 1.

			[1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       8        1
*      64       24        1
*     512      448       48       1
*    4096     7680     1600      80      1
*   32768   126976    46080    4160    120     1
*  262144  2064384  1232896  179200   8960   168   1
* 2097152 33292288 31653888 6967296 537600 17024 224 1
(End)

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

Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507.

Cf. A075502, A075504.

Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (8^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*8)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 8m*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}(1-8k*x), m >= 1.
E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1.

A075508 Shifts one place left under 9th-order binomial transform.

Original entry on oeis.org

1, 1, 10, 109, 1351, 19612, 333451, 6493069, 141264820, 3376695763, 87799365343, 2465959810690, 74353064138749, 2393123710957813, 81812390963020066, 2958191064076428793, 112727516544416978299, 4513118224822056822772, 189305466502867876489519

Offset: 0

Wolfdieter Lang, Oct 02 2002

Previous name was: Row sums of triangle A075504 (for n>=1).

Muniru A Asiru, Table of n, a(n) for n = 0..108

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

List([0..20],n->Sum([0..n],m->9^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018

[seq(factorial(k)*coeftayl(exp((exp(9*x)-1)/9), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018

Table[9^n BellB[n, 1/9], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n) = Sum_{m=0..n} 9^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).
E.g.f.: exp((exp(9*x)-1)/9).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 9*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 9^n * n^n * exp(n/LambertW(9*n) - 1/9 - n) / (sqrt(1 + LambertW(9*n)) * LambertW(9*n)^n). - Vaclav Kotesovec, Jul 15 2021

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015

A075505 Stirling2 triangle with scaled diagonals (powers of 10).

Original entry on oeis.org

1, 10, 1, 100, 30, 1, 1000, 700, 60, 1, 10000, 15000, 2500, 100, 1, 100000, 310000, 90000, 6500, 150, 1, 1000000, 6300000, 3010000, 350000, 14000, 210, 1, 10000000, 127000000, 96600000, 17010000, 1050000, 26600, 280, 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(10*z) - 1)*x/10) - 1.

			[1]; [10,1]; [100,30,1]; ...; p(3,x) = x(100 + 30*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*        1
*       10         1
*      100        30        1
*     1000       700       60        1
*    10000     15000     2500      100       1
*   100000    310000    90000     6500     150     1
*  1000000   6300000  3010000   350000   14000   210   1
* 10000000 127000000 96600000 17010000 1050000 26600 280 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.

Row sums are A075509.

Cf. A075504.

Flatten[Table[10^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 11, for(m=1, n, print1(10^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

a(n, m) = (10^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*10)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 10m*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} (1-10k*x), m >= 1.
E.g.f. for m-th column: (((exp(10x)-1)/10)^m)/m!, m >= 1.

cp's OEIS Frontend

A076013 Seventh column of triangle A075504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A076008 Second column of triangle A075504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A076009 Third column of triangle A075504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A076010 Fourth column of triangle A075504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A076011 Fifth column of triangle A075504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A076012 Sixth column of triangle A075504.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A075503 Stirling2 triangle with scaled diagonals (powers of 8).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A075508 Shifts one place left under 9th-order binomial transform.

Views

Author

Keywords

Comments

Links

Crossrefs