A075513 -id:A075513 - cp's OEIS frontend

A076012 Sixth column of triangle A075504.

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!.

A154715 Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).

Original entry on oeis.org

1, 2, 3, 4, 18, 16, 8, 81, 192, 125, 16, 324, 1536, 2500, 1296, 32, 1215, 10240, 31250, 38880, 16807, 64, 4374, 61440, 312500, 699840, 705894, 262144, 128, 15309, 344064, 2734375, 9797760, 17294403, 14680064, 4782969

Offset: 0

Peter Luschny, Jan 14 2009

Formatted as a square array:
1st row is A000079(n). Subsets of an n-set.
2nd row is A036290(n+1). Special (n+1)-subsets of a 3n-set partitioned into 3-blocks.
2nd column is A066274(n+1). Endofunctions of [n] such that 1 is not a fixed point.
1st column is A000272(n+2). Trees on n labeled nodes (Cayley's formula).
Alternating sum of rows in the triangle, Sum_{k=0..n} (-1)^(n-k) * T(n,k) = n! = A000142(n).
This triangle gives the coefficient of Sidi's polynomials D_{n,2,n}(-z)/(-z), for n >= 0. See [Sidi 1980]. - Wolfdieter Lang, Oct 27 2022

			Triangle begins as:
   1;
   2,    3;
   4,   18,    16;
   8,   81,   192,    125;
  16,  324,  1536,   2500,   1296;
  32, 1215, 10240,  31250,  38880,  16807;
  64, 4374, 61440, 312500, 699840, 705894, 262144;

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874. Eq. (4.10), p. 862.

Cf. A000079, A000272, A036290, A066274, A075513.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*(k+2)^n ))); # G. C. Greubel, May 09 2019

[[Binomial(n,k)*(k+2)^n: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 09 2019

T := proc(n,k) binomial(n,k)*(k+2)^n end;

Table[Binomial[n, k]*(k+2)^n, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 09 2019 *)

{T(n, k) = binomial(n,k)*(k+2)^n}; \\ G. C. Greubel, May 09 2019

[[binomial(n,k)*(k+2)^n for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 09 2019

T(n,k) = binomial(n,k)*(k+2)^n, where n >= 0, and k >= 0.
From Wolfdieter Lang, Oct 20 2022: (Start)
O.g.f. of column k: (-x)^k*(k + 2)^k/(1 - (k + 2)*x)^(k+1), for k >= 0. See |A075513| with offset 0.
E.g.f. of column k: exp((k+2)*x)*((k+2)*x)^k/k!, for k >= 0. (End)
E.g.f. of triangle (of row polynomials in y): exp(2*x)*substitute(z = x*y*exp(x), LambertW(-z)^2/(-z)*2*(1 + LambertW(-z)))). - Wolfdieter Lang, Oct 24 2022

A075908 Fifth column of triangle A075499.

Original entry on oeis.org

1, 60, 2240, 67200, 1779456, 43545600, 1010606080, 22600089600, 492077121536, 10505429975040, 221005133905920, 4597756408627200, 94837435443183616, 1943344895628410880, 39618196941842677760

Offset: 0

Wolfdieter Lang, Oct 02 2002

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

Cf. A075907, A075909.

a(n) = A075499(n+5, 5) = (4^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} A075513(5, m)*((m+1)*4)^n/4!.
G.f.: 1/Product_{k=1..5} (1 - 4*k*x).
E.g.f.: (d^5/dx^5)(((exp(4*x)-1)/4)^5)/5! = (exp(4*x) - 64*exp(8*x) + 486*exp(12*x) - 1024*exp(16*x) + 625*exp(20*x))/4!.

A075910 Seventh column of triangle A075499.

Original entry on oeis.org

1, 112, 7392, 376320, 16380672, 642453504, 23410376704, 808210923520, 26787271999488, 860325833342976, 26956901684084736, 828217683974553600, 25047119070415028224, 747831252926309859328, 22095179333791056396288

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..6} A075513(7,m)*exp(4*(m+1)*x)/6!.

Cf. A075509.

CoefficientList[Series[1/Product[1-4k x,{k,7}],{x,0,20}],x] (* Harvey P. Dale, Aug 11 2021 *)

a(n) = A075499(n+7, 7) = (4^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} A075513(7, m)*((m+1)*4)^n/6!.
G.f.: 1/Product_{k=1..7} (1 - 4*k*x).
E.g.f.: (d^7/dx^7)(((exp(4*x)-1)/4)^7)/7! = (exp(4*x) - 384*exp(8*x) + 10935*exp(12*x) - 81920*exp(16*x) + 234375*exp(20*x) - 279936*exp(24*x) + 117649*exp(28*x))/6!.

A075917 Fourth column of triangle A075501.

Original entry on oeis.org

1, 60, 2340, 75600, 2204496, 60419520, 1591202880, 40800672000, 1027086863616, 25522067450880, 628349082117120, 15366613964083200, 373968813041012736, 9068526888588656640, 219326169845571010560

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..3} A075513(4,m)*exp(6*(m+1)*x)/3!.

Cf. A075916, A075918.

a(n) = A075501(n+4, 4) = (6^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..3} A075513(4, m)*((m+1)*6)^n/3!.
G.f.: 1/Product_{k=1..4} (1 - 6*k*x).
E.g.f.: (d^4/dx^4)(((exp(6*x)-1)/6)^4)/4! = (-exp(6*x) + 24*exp(12*x) - 81*exp(18*x) + 64*exp(24*x))/3!.

A075918 Fifth column of triangle A075501.

Original entry on oeis.org

1, 90, 5040, 226800, 9008496, 330674400, 11511434880, 386143718400, 12611398415616, 403864019919360, 12744269679697920, 397694704355020800, 12304809943691636736, 378212825199337758720, 11565710925825703772160

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..4} A075513(5,m)*exp(6*(m+1)*x)/4!.

Cf. A075917, A075919.

a(n) = A075501(n+5, 5) = (6^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} A075513(5, m)*((m+1)*6)^n/4!.
G.f.: 1/Product_{k=1..5} (1 - 6*k*x).
E.g.f.: (d^5/dx^5)(((exp(6*x)-1)/6)^5)/5! = (exp(6*x) - 64*exp(12*x) + 486*exp(18*x) - 1024*exp(24*x) + 625*exp(30*x))/4!.

A075919 Sixth column of triangle A075501.

Original entry on oeis.org

1, 126, 9576, 571536, 29583792, 1395690912, 61756307712, 2609370796032, 106548747072768, 4239618914539008, 165370550603102208, 6351034526066700288, 240942052882092847104, 9052126728954680254464

Offset: 0

Wolfdieter Lang, Oct 02 2002

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

Cf. A075918, A075920.

a(n) = A075501(n+6, 6) = (6^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} A075513(6, m)*((m+1)*6)^n/5!.
G.f.: 1/Product_{k=1..6} (1 - 6*k*x).
E.g.f.: (d^6/dx^6)(((exp(6*x)-1)/6)^6)/6! = (-exp(6*x) + 160*exp(12*x) - 2430*exp(18*x) + 10240*exp(24*x) - 15625*exp(30*x) + 7776*exp(36*x))/5!.

A075920 Seventh column of triangle A075501.

Original entry on oeis.org

1, 168, 16632, 1270080, 82927152, 4878631296, 266658822144, 13809041326080, 686528482768128, 33073815190800384, 1554470788616718336, 71638807647968870400, 3249771974096785403904, 145542549641019667218432

Offset: 0

Wolfdieter Lang, Oct 02 2002

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

Cf. A075919.

a(n) = A075501(n+7, 7) = (6^n)S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} A075513(7, m)*((m+1)*6)^n/6!.
G.f.: 1/Product_{k=1..7} (1 - 6*k*x).
E.g.f.: (d^7/dx^7)(((exp(6*x)-1)/6)^7)/7! = (exp(6*x) - 384*exp(12*x) + 10935*exp(18*x) - 81920*exp(24*x) + 234375*exp(30x) - 279936*exp(36*x) + 117649*exp(42*x))/6!.

A075921 Second column of triangle A075502.

Original entry on oeis.org

1, 21, 343, 5145, 74431, 1058841, 14941423, 210003465, 2945813311, 41281739961, 578226834703, 8097153012585, 113373983463391, 1587332657497881, 22223335428043183, 311131443554114505

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..1} A075513(2,m)*exp(7*(m+1)*x).

Index entries for linear recurrences with constant coefficients, signature (21, -98).

Cf. A000420 (first column), A075922.

Table[-7^n+2 14^n,{n,0,20}] (* or *) LinearRecurrence[{21,-98}, {1,21},20] (* Harvey P. Dale, Apr 30 2011 *)

a(n) = A075502(n+2, 2) = (7^n)*S2(n+2, 2) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = -7^n + 2*14^n.
G.f.: 1/((1-7*x)*(1-14*x)).
E.g.f.: (d^2/dx^2)(((exp(7*x)-1)/7)^2)/2! = -exp(7*x) + 2*exp(14*x).
a(0)=1, a(1)=21, a(n) = 21a(n-1) - 98a(n-2). - Harvey P. Dale, Apr 30 2011

A075922 Third column of triangle A075502.

Original entry on oeis.org

1, 42, 1225, 30870, 722701, 16235562, 355888225, 7683656190, 164302593301, 3491636199282, 73902587019625, 1560051480424710, 32874455072382301, 691950889177526202, 14553192008156093425, 305928163614832076430

Offset: 0

Wolfdieter Lang, Oct 02 2002

The e.g.f. given below is Sum_{m=0..2} A075513(3,m)*exp(7*(m+1)*x)/2!.

Cf. A075921, A075923.

a(n) = A075502(n+3, 3) = (7^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (7^n - 8*14^n + 9*21^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 7*k*x).
E.g.f.: (d^3/dx^3)(((exp(7*x)-1)/7)^3)/3! = (exp(7*x) - 8*exp(14*x) + 9*exp(21*x))/2!.

cp's OEIS Frontend

A076012 Sixth column of triangle A075504.

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A154715 Triangle interpolating between the subsets of an n-set (A000079) and the trees on n labeled nodes (A000272) (read by rows).

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A075908 Fifth column of triangle A075499.

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A075910 Seventh column of triangle A075499.

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A075917 Fourth column of triangle A075501.

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A075918 Fifth column of triangle A075501.

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A075919 Sixth column of triangle A075501.

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A075920 Seventh column of triangle A075501.

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A075921 Second column of triangle A075502.

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