A036226 -id:A036226 - cp's OEIS frontend

A053110 Expansion of (-1 + 1/(1-7x)^7)/(49x); related to A036226.

1, 28, 588, 10290, 158466, 2218524, 28840812, 353299947, 4121832715, 46164526408, 499416240232, 5243870522436, 53648829191076, 536488291910760, 5257585260725448, 50604258134482437, 479252091744216021

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (49,-1029,12005,-84035,352947,-823543,823543).

Cf. A036226, A001792, A036083, A036224.

[7^(n-1)*Binomial(n+7, 6): n in [0..30]]; // G. C. Greubel, Aug 16 2018

CoefficientList[Series[(-1+1/(1-7x)^7)/(49x),{x,0,30}],x] (* or *) LinearRecurrence[{49,-1029,12005,-84035,352947,-823543,823543},{1,28,588,10290,158466,2218524,28840812},30] (* Harvey P. Dale, Jun 03 2015 *)
Table[7^(n-1)*Binomial[n+7, 6], {n,0,30}] (* G. C. Greubel, Aug 16 2018 *)

vector(30,n,n--; 7^(n-1)*binomial(n+7, 6)) \\ G. C. Greubel, Aug 16 2018

[lucas_number2(n, 7, 0)*binomial(n,6)/7^8 for n in range(7, 24)] # Zerinvary Lajos, Mar 13 2009

a(n) = 7^(n-1)*binomial(n+7, 6);

G.f.: (-1 + (1-7*x)^(-7))/(x*7^2).

A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.

Original entry on oeis.org

1, -1, 2, 1, -8, 9, -1, 24, -81, 64, 1, -64, 486, -1024, 625, -1, 160, -2430, 10240, -15625, 7776, 1, -384, 10935, -81920, 234375, -279936, 117649, -1, 896, -45927, 573440, -2734375, 5878656, -5764801, 2097152, 1, -2048, 183708, -3670016, 27343750, -94058496, 161414428, -134217728, 43046721

Offset: 1

Wolfdieter Lang, Oct 02 2002

Coefficients of the Sidi polynomials (-1)^(n-1)*D_{n-1,1,n-1}(x), for n >=1, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980].
The row polynomials p(n, x) := Sum_{m=0..n-1} a(n, m)x^m, n >= 1, are obtained from ((Eu(x)^n)*(x-1)^n)/(n*x), where Eu(x) := xd/dx is the Euler-derivative with respect to x.
The row polynomials p(n, y) := Sum_{m=0..n-1} a(n, m)*y^m, n >= 1, are also obtained from ((d^m/dx^m)((exp(x)-1)^m)/m)/exp(x) after replacement of exp(x) by y. Here (d^m/dx^m)f(x), m >= 1, denotes m-fold differentiation of f(x) with respect to x.
b(k,m,n) := (Sum_{p=0..m-1} (a(m, p)*((p+1)*k)^n))/(m-1)!, n >= 0, has g.f. 1/Product_{p=1..m} (1 - k*p*x) for k = 1, 2,... and m = 1, 2,...
The (signed) row sums give A000142(n-1), n >= 1, (factorials) and (unsigned) A074932(n).
The (unsigned) columns give A000012 (powers of 1), 2*A001787(n+1), (3^2)*A027472(n), (4^3)*A038846(n-1), (5^4)*A036071(n-5), (6^5)*A036084(n-6), (7^6)*A036226(n-7), (8^7)*A053107(n-8) for m=0..7.
Right edge of triangle is A000169. - Michel Marcus, May 17 2013

			The triangle T(n, m)  begins:
  n\m 0     1      2        3        4         5         6          7       8
  1:  1
  2: -1     2
  3:  1    -8      9
  4: -1    24    -81       64
  5:  1   -64    486    -1024      625
  6: -1   160  -2430    10240   -15625      7776
  7:  1  -384  10935   -81920   234375   -279936    117649
  8: -1   896 -45927   573440 -2734375   5878656  -5764801    2097152
  9:  1 -2048 183708 -3670016 27343750 -94058496 161414428 -134217728 4304672
  [Reformatted by _Wolfdieter Lang_, Oct 12 2022]
-----------------------------------------------------------------------------
p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.

A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003.

Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Doron S. Lubinsky and Herbert Stahl, Some Explicit Biorthogonal Polynomials, (IN) Approximation Theory XI, (C.K. Chui, M. Neamtu, L. Schumaker, eds.), Nashboro Press, Nashville, 2005, pp. 279-285.
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.

Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013, A258773.

# Assuming offset 0.
seq(seq((-1)^(n-k)*binomial(n, k)*(k+1)^n, k=0..n), n=0..8);
# Alternative:
egf := x -> 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y):
ser := x -> series(egf(x), x, 12):
row := n -> seq(coeff(n!*coeff(ser(x), x, n), y, k), k=0..n):
seq(print(row(n)), n = 0..8); # Peter Luschny, Oct 21 2022

p[n_, x_] := p[n, x] = Nest[ x*D[#, x]& , (x-1)^n, n]/(n*x); a[n_, m_] := Coefficient[ p[n, x], x, m]; Table[a[n, m], {n, 1, 9}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 03 2013 *)

tabl(nn) = {for (n=1, nn, for (m=0, n-1, print1((-1)^(n-m-1)*binomial(n-1, m)*(m+1)^(n-1), ", ");); print(););} \\ Michel Marcus, May 17 2013

T(n, m) = ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n >= m+1 >= 1, else 0.
G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m >= 0.
E.g.f.: -LambertW(-x*y*exp(-x))/((1+LambertW(-x*y*exp(-x)))*x*y). - Vladeta Jovovic, Feb 13 2008 [corrected for offset 0 <= m <= n. For offset n >= 1 take the integral over x. - Wolfdieter Lang, Oct 12 2022]
T(n, k) = S(n, k+1) / n where S(, ) is triangle in A258773. - Michael Somos, May 13 2018
E.g.f. of column k, with offset n >= 0: exp(-(k + 1)*x)*((k + 1)*x)^k/k!. - Wolfdieter Lang, Oct 20 2022
E.g.f: 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y) assuming offset = 0. - Peter Luschny, Oct 21 2022

A053107 Expansion of 1/(1-8*x)^8.

Original entry on oeis.org

1, 64, 2304, 61440, 1351680, 25952256, 449839104, 7197425664, 107961384960, 1535450808320, 20882130993152, 273366078455808, 3462636993773568, 42617070692597760, 511404848311173120, 6000483553517764608, 69005560865454292992, 779356922715719073792

Offset: 0

Wolfdieter Lang

With a different offset, number of n-permutations (n>=7) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly 7 u's. - Zerinvary Lajos, Feb 11 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (64, -1792, 28672, -286720, 1835008, -7340032, 16777216, -16777216).

Cf. A036226.

Cf. A081138, A140802, A175210, A140406, A053107, A141054, A173155. - Zerinvary Lajos, Feb 11 2010

[8^n* Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Table[Binomial[n + 7, 7]*8^n, {n, 0, 20}] (* Zerinvary Lajos, Feb 11 2010 *)
CoefficientList[Series[1/(1-8x)^8,{x,0,20}],x] (* or *) LinearRecurrence[ {64,-1792,28672,-286720,1835008,-7340032,16777216,-16777216},{1,64,2304,61440,1351680,25952256,449839104,7197425664},20] (* Harvey P. Dale, Jul 19 2018 *)

vector(30, n, n--; 8^n*binomial(n+7,7)) \\ G. C. Greubel, Aug 16 2018

[lucas_number2(n, 8, 0)*binomial(n,7)/8^7 for n in range(7, 22)] # Zerinvary Lajos, Mar 13 2009

a(n) = 8^n*binomial(n+7, 7).

G.f.: 1/(1-8*x)^8.

More terms from Harvey P. Dale, Jul 19 2018

A140406 a(n) = binomial(n+6, 6)*8^n.

Original entry on oeis.org

1, 56, 1792, 43008, 860160, 15138816, 242221056, 3598712832, 50381979648, 671759728640, 8598524526592, 106309030510592, 1275708366127104, 14915974742409216, 170468282770391040, 1909244767028379648, 21001692437312176128, 227312435792084729856

Offset: 0

Zerinvary Lajos, Jun 16 2008

With a different offset, number of n-permutations (n >= 6) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly six (6) u's.
If n=6 then a(0)=1.
Example: a(1)=56 because we have
uuuuuup, uuuuupu, uuuupuu, uuupuuu, uupuuuu, upuuuuu, puuuuuu,
uuuuuur, uuuuuru, uuuuruu, uuuruuu, uuruuuu, uruuuuu, ruuuuuu,
uuuuuus, uuuuusu, uuuusuu, uuusuuu, uusuuuu, usuuuuu, suuuuuu,
uuuuuut, uuuuutu, uuuutuu, uuutuuu, uutuuuu, utuuuuu, tuuuuuu,
uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu,
uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu,
uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu,
uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (56,-1344,17920,-143360,688128,-1835008,2097152).

Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140405.

[8^n* Binomial(n+6, 6): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Maple
```
seq(binomial(n+6,6)*8^n,n=0..17);
```

Table[Binomial[n+6,6]8^n,{n,0,20}] (* or *) LinearRecurrence[ {56,-1344,17920,-143360,688128,-1835008,2097152},{1,56,1792,43008,860160,15138816,242221056},20] (* Harvey P. Dale, Dec 15 2011 *)

a(n)=binomial(n+6,6)<<(3*n) \\ Charles R Greathouse IV, Dec 15 2011

G.f.: 1/(1-8*x)^7. - Zerinvary Lajos, Aug 06 2008
a(n) = 56*a(n-1) - 1344*a(n-2) + 17920*a(n-3) - 143360*a(n-4) + 688128*a(n-5) - 1835008*a(n-6) + 2097152*a(n-7). - Harvey P. Dale, Dec 15 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 538628/5 - 806736*log(8/7).
Sum_{n>=0} (-1)^n/a(n) = 2834352*log(9/8) - 1669188/5. (End)

A053109 Expansion of 1/(1-10*x)^10.

Original entry on oeis.org

1, 100, 5500, 220000, 7150000, 200200000, 5005000000, 114400000000, 2431000000000, 48620000000000, 923780000000000, 16796000000000000, 293930000000000000, 4974200000000000000, 81719000000000000000

Offset: 0

Wolfdieter Lang

This is the tenth member of the k-family of sequences a(k,n) := k^n*binomial(n+k-1,k-1) starting with A000012 (powers of 1), A001787(n+1), A027472(n+3), A038846, A036071, A036084, A036226, A053107-9 for k=1..10.

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (100, -4500, 120000, -2100000, 25200000, -210000000, 1200000000, -4500000000, 10000000000, -10000000000).

List([0..15],n->10^n*Binomial(n+9,9)); # Muniru A Asiru, Aug 16 2018

[10^n*Binomial(n+9, 9): n in [0..30]]; // G. C. Greubel, Aug 16 2018

seq(coeff(series(1/(1-10*x)^10, x, n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Aug 16 2018

CoefficientList[Series[1/(1-10x)^10,{x,0,20}],x] (* or *) Table[10^n Binomial[n+9,9],{n,0,20}] (* Harvey P. Dale, May 19 2011 *)

vector(30,n,n--; 10^n*binomial(n+9, 9)) \\ G. C. Greubel, Aug 16 2018

[lucas_number2(n, 10, 0)*binomial(n,9)/10 ^9 for n in range(9, 24)] # Zerinvary Lajos, Mar 13 2009

a(n) = 10^n*binomial(n+9, 9);

G.f.: 1/(1-10*x)^10.

A170932 a(n) = binomial(n + 8, 8)*7^n .

Original entry on oeis.org

1, 63, 2205, 56595, 1188495, 21630609, 353299947, 5299499205, 74192988870, 980996186170, 12360551945742, 149450309889426, 1743586948709970, 19715944727720430, 216875392004924730, 2327795874186192102, 24441856678955017071, 251607348165713411025

Offset: 0

Zerinvary Lajos, Feb 08 2010

With a different offset, number of n-permutations of 8 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight, (8) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (63,-1764,28812,-302526,2117682,-9882516,29647548,-51883209,40353607).

Cf. A027474, A140107, A139641, A140404, A036226, A050989.

[Binomial(n + 8, 8)*7^n: n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 8, 8]*7^n, {n, 0, 20}]

a(n) = C(n + 8, 8)*7^n.
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 12082656/5 - 15676416*log(7/6).
Sum_{n>=0} (-1)^n/a(n) = 117440512*log(8/7) - 235229912/15. (End)

A140405 a(n) = binomial(n+6, 6)*5^n.

Original entry on oeis.org

1, 35, 700, 10500, 131250, 1443750, 14437500, 134062500, 1173046875, 9775390625, 78203125000, 604296875000, 4532226562500, 33120117187500, 236572265625000, 1656005859375000, 11385040283203125, 77016448974609375, 513442993164062500, 3377914428710937500

Offset: 0

Zerinvary Lajos, Jun 16 2008

With a different offset, number of n-permutations (n>=6) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly six (6) u's.
If n=6 then a(0)=1.
Example: a(1)=35 because we have
uuuuuut, uuuuutu, uuuutuu, uuutuuu, uutuuuu, utuuuuu, tuuuuuu,
uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu,
uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu,
uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu,
uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu.

Index entries for linear recurrences with constant coefficients, signature (35,-525,4375,-21875,65625,-109375,78125).

Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140406.

Maple
```
seq(binomial(n+6,6)*5^n,n=0..18);
```

Table[Binomial[n+6,6]5^n,{n,0,20}] (* Harvey P. Dale, Dec 03 2017 *)

G.f.: 1/(1-5*x)^7. - Zerinvary Lajos, Aug 06 2008
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 6856 - 30720*log(5/4).
Sum_{n>=0} (-1)^n/a(n) = 233280*log(6/5) - 42531. (End)

A293270 a(n) = n^nbinomial(2n-1, n).

Original entry on oeis.org

1, 1, 12, 270, 8960, 393750, 21555072, 1413199788, 107961384960, 9418192087590, 923780000000000, 100633991211229476, 12055263261877075968, 1575041416811693275900, 222887966509090352332800, 33962507149515380859375000, 5543988061027763016035205120

Offset: 0

Ilya Gutkovskiy, Oct 04 2017

nonn

The n-th term of the n-fold convolution of the powers of n.

Cf. A000312, A001700, A088218.

Cf. A001787, A027472, A036071, A036084, A036226, A038846, A053107, A053108, A053109.

Join[{1}, Table[n^n Binomial[2 n - 1, n], {n, 1, 16}]]
Join[{1}, Table[(-1)^n n^n Binomial[-n, n], {n, 1, 16}]]
Table[SeriesCoefficient[1/(1 - n x)^n, {x, 0, n}], {n, 0, 16}]

a(n) = n^n*binomial(2*n-1, n); \\ Altug Alkan, Oct 04 2017

a(n) = [x^n] 1/(1 - n*x)^n.

a(n) ~ 2^(2*n-1)*n^n/sqrt(Pi*n).

cp's OEIS Frontend

A053110 Expansion of (-1 + 1/(1-7*x)^7)/(49*x); related to A036226.

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A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.

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A053107 Expansion of 1/(1-8*x)^8.

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A140406 a(n) = binomial(n+6, 6)*8^n.

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A053109 Expansion of 1/(1-10*x)^10.

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A170932 a(n) = binomial(n + 8, 8)*7^n .

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A140405 a(n) = binomial(n+6, 6)*5^n.

A053110 Expansion of (-1 + 1/(1-7x)^7)/(49x); related to A036226.

A293270 a(n) = n^nbinomial(2n-1, n).