A269953 -id:A269953 - cp's OEIS frontend

A001113 Decimal expansion of e.

2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, 0, 4, 5, 2, 3, 5, 3, 6, 0, 2, 8, 7, 4, 7, 1, 3, 5, 2, 6, 6, 2, 4, 9, 7, 7, 5, 7, 2, 4, 7, 0, 9, 3, 6, 9, 9, 9, 5, 9, 5, 7, 4, 9, 6, 6, 9, 6, 7, 6, 2, 7, 7, 2, 4, 0, 7, 6, 6, 3, 0, 3, 5, 3, 5, 4, 7, 5, 9, 4, 5, 7, 1, 3, 8, 2, 1, 7, 8, 5, 2, 5, 1, 6, 6, 4, 2, 7, 4, 2, 7, 4, 6

Offset: 1

N. J. A. Sloane

e is sometimes called Euler's number or Napier's constant.
Also, decimal expansion of sinh(1)+cosh(1). - Mohammad K. Azarian, Aug 15 2006
If m and n are any integers with n > 1, then |e - m/n| > 1/(S(n)+1)!, where S(n) = A002034(n) is the smallest number such that n divides S(n)!. - Jonathan Sondow, Sep 04 2006
Limit_{n->infinity} A000166(n)*e - A000142(n) = 0. - Seiichi Kirikami, Oct 12 2011
Euler's constant (also known as Euler-Mascheroni constant) is gamma = 0.57721... and Euler's number is e = 2.71828... . - Mohammad K. Azarian, Dec 29 2011
One of the many continued fraction expressions for e is 2+2/(2+3/(3+4/(4+5/(5+6/(6+ ... from Ramanujan (1887-1920). - Robert G. Wilson v, Jul 16 2012
e maximizes the value of x^(c/x) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. This explains why elements of A000792 are composed primarily of factors of 3, and where needed, some factors of 2. These are the two primes closest to e. - Richard R. Forberg, Oct 19 2014
There are two real solutions x to c^x = x^c when c, x > 0 and c != e, one of which is x = c, and only one real solution when c = e, where the solution is x = e. - Richard R. Forberg, Oct 22 2014
This is the expected value of the number of real numbers that are independently and uniformly chosen at random from the interval (0, 1) until their sum exceeds 1 (Bush, 1961). - Amiram Eldar, Jul 21 2020

			2.71828182845904523536028747135266249775724709369995957496696762772407663...

Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 400.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 250-256.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers, p. 85.
E. Maor, e: The Story of a Number, Princeton Univ. Press, 1994.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 52.
G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places. Math. Tables and Other Aids to Computation 4, (1950). 11-15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 1 and 2, equations 1:7:4, 2:5:4 at pages 13, 20.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.

N. J. A. Sloane, Table of 50000 digits of e labeled from 1 to 50000 [based on the ICON Project link below]
Mohammad K. Azarian, An Expansion of e, Problem # B-765, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 4, Aug. 1995, p. 377.
Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.
L. E. Bush, The William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 68, No. 1 (1961), pp. 18-33, problem 3.
Ed Copeland and Brady Haran, A proof that e is irrational, Numberphile video (2021).
Dave's Math Tables, e
X. Gourdon, Plouffe's Inverter, e to 1.250 billion digits
X. Gourdon and P. Sebah, The constant e and its computation
Brady Haran and James Grime, Incredible Formula, Numberphile YouTube video, 2016.
ICON Project, e to 50000 places
Roger Mansuy, Un intrigant poème... mathématique, Images des Mathématiques, CNRS, 2023. In French.
R. Nemiroff and J. Bonnell, The first 5 million digits of the number e
Remco Niemeijer, Digits Of E, programmingpraxis
J. J. O'Connor & E. F. Robertson, The number e
Michael Penn, e is irrational, YouTube video, 2020.
Simon Plouffe, A million digits
G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places, Pi, A Source book, pp 277-281, 2000.
E. Sandifer, How Euler Did It, Who proved e is irrational?, MAA Online (2006)
D. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, Math. Comp., 23 (1969), 679-680.
Jean-Louis Sigrist, Le premier million de décimales de e.
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 339.
Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641 (article) and 114 (2007), 659 (addendum).
Jonathan Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
G. Villemin's Almanach of Numbers, Constant "e"
Eric Weisstein's World of Mathematics, e
Eric Weisstein's World of Mathematics, e Digits
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Uniform Sum Distribution
Eric Weisstein's World of Mathematics, e Approximations
Wikipedia, E (mathematical constant)
Index entries for "core" sequences
Index entries for transcendental numbers

Cf. A002034, A003417 (continued fraction), A073229, A122214, A122215, A122216, A122217, A122416, A122417.
Expansion of e in base b: A004593 (b=2), A004594 (b=3), A004595 (b=4), A004596 (b=5), A004597 (b=6), A004598 (b=7), A004599 (b=8), A004600 (b=9), this sequence (b=10), A170873 (b=16). - Jason Kimberley, Dec 05 2012
Powers e^k: A092578 (k = -7), A092577 (k = -6), A092560 (k = -5), A092553 - A092555 (k = -2 to -4), A068985 (k = -1), A072334 (k = 2), A091933 (k = 3), A092426 (k = 4), A092511 - A092513 (k = 5 to 7).

-- See Niemeijer link.
a001113 n = a001113_list !! (n-1)
a001113_list = eStream (1, 0, 1)
   [(n, a * d, d) | (n, d, a) <- map (\k -> (1, k, 1)) [1..]] where
   eStream z xs'@(x:xs)
     | lb /= approx z 2 = eStream (mult z x) xs
     | otherwise = lb : eStream (mult (10, -10 * lb, 1) z) xs'
     where lb = approx z 1
           approx (a, b, c) n = div (a * n + b) c
           mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)
-- Reinhard Zumkeller, Jun 12 2013

Digits := 200: it := evalf((exp(1))/10, 200): for i from 1 to 200 do printf(`%d,`,floor(10*it)): it := 10*it-floor(10*it): od: # James Sellers, Feb 13 2001

RealDigits[E, 10, 120][[1]] (* Harvey P. Dale, Nov 14 2011 *)

e = Sum_{k >= 0} 1/k! = lim_{x -> 0} (1+x)^(1/x).
e is the unique positive root of the equation Integral_{u = 1..x} du/u = 1.
exp(1) = ((16/31)*(1 + Sum_{n>=1} ((1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!)))^2. Robert Israel confirmed that the above formula is correct, saying: "In fact, Sum_{n=0..oo} n^j*t^n/n! = P_j(t)*exp(t) where P_0(t) = 1 and for j >= 1, P_j(t) = t (P_(j-1)'(t) + P_(j-1)(t)). Your sum is 1/2*P_3(1/2) + 1/2*P_1(1/2) + P_0(1/2)." - Alexander R. Povolotsky, Jan 04 2009
exp(1) = (1 + Sum_{n>=1} ((1+n+n^3)/n!))/7. - Alexander R. Povolotsky, Sep 14 2011
e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2. - Rok Cestnik, Jan 19 2017
From Peter Bala, Nov 13 2019: (Start)
The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,..., where R(n,x) is the n-th row polynomial of A269953.
e = 2 + Sum_{n >= 0} (-1)^n*(n+2)!/(d(n+2)*d(n+3)), where d(n) = A000166(n).
e = Sum_{n >= 0} (x^2 + (n+2)*x + n)/(n!(n + x)*(n + 1 + x)), provided x is not zero or a negative integer. (End)
Equals lim_{n -> oo} (2*3*5*...*prime(n))^(1/prime(n)). - Peter Luschny, May 21 2020
e = 3 - Sum_{n >= 0} 1/((n+1)^2*(n+2)^2*n!). - Peter Bala, Jan 13 2022
e = lim_{n->oo} prime(n)*(1 - 1/n)^prime(n). - Thomas Ordowski, Jan 31 2023
e = 1+(1/1)*(1+(1/2)*(1+(1/3)*(1+(1/4)*(1+(1/5)*(1+(1/6)*(...)))))), equivalent to the first formula. - David Ulgenes, Dec 01 2023
From Michal Paulovic, Dec 12 2023: (Start)
Equals lim_{n->oo} (1 + 1/n)^n.
Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End)
Equals Product_{k>=1} (1 + 1/k) * (1 - 1/(k + 1)^2)^k. - Antonio Graciá Llorente, May 14 2024
Equals lim_{n->oo} Product_{k=1..n} (n^2 + k)/(n^2 - k) (see Finch). - Stefano Spezia, Oct 19 2024
e ~ (1 + 9^((-4)^(7*6)))^(3^(2^85)), correct to more than 18*10^24 digits (Richard Sabey, 2004); see Haran and Grime link. - Paolo Xausa, Dec 21 2024.

A105793 Expansion of e.g.f. (1 + y)^(1 + x).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, -1, 0, 1, 0, 2, -1, -2, 1, 0, -6, 5, 5, -5, 1, 0, 24, -26, -15, 25, -9, 1, 0, -120, 154, 49, -140, 70, -14, 1, 0, 720, -1044, -140, 889, -560, 154, -20, 1, 0, -5040, 8028, -64, -6363, 4809, -1638, 294, -27, 1, 0, 40320, -69264, 8540, 50840, -44835, 17913, -3990, 510, -35, 1

Offset: 0

Paul Barry, Apr 20 2005

Generalized Stirling number triangle of first kind. Row sums are (1,2,2,0,0,0,...) = 2C(2,n) - 2C(1,n) + C(0,n). Inverse is A105794.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, -1, 0, -2, -1, -3, -2, -4, -3, -5, -4, -6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 23 2006

			From _Wolfdieter Lang_, Jun 19 2017: (Start)
The triangle T(n, k) starts
  n\k  0     1      2    3     4      5     6     7   8   9 10 ...
  0:   1
  1:   1     1
  2:   0     1      1
  3:   0    -1      0    1
  4:   0     2     -1   -2     1
  5:   0    -6      5    5    -5      1
  6:   0    24    -26  -15    25     -9     1
  7:   0  -120    154   49  -140     70   -14     1
  8:   0   720  -1044 -140   889   -560   154   -20   1
  9:   0 -5040   8028  -64 -6363   4809 -1638   294 -27   1
  10:  0 40320 -69264 8540 50840 -44835 17913 -3990 510 -35  1
  ... reformatted
Recurrence from a- and z-sequence (see above): T(1, 0) = T(0, 0) = 1; T(1, 1) = (1/1)*(1*T(0, 0)) = 1, T(2, 0) = 2*(T(1, 0) - T(1, 1)) = 0, T(2, 1) = (2/1)*(T(1,0) + (-1/2)*T(1, 1)) = 1. T(3, 1) = (3/1)*(0 + (-1/2)*T(2, 1) + (1/6)*T(2, 2)) = -1. (End)

Cf. A007318, A027641/A027642, A033999, A048994, A094645, A269953.

t[0, 0] = 1; t[n_, k_] := StirlingS1[n, k] + n*StirlingS1[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 04 2013, after Giuliano Cabrele *)

E.g.f.: (1+y)^(1+x); rows have g.f. k!*binomial(x+1, k); Columns have g.f. (1+x)*log(1+x)^k.
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then T(n,i) = f(n,i,-1), for n=1,2,...; i=0...n. - Milan Janjic, Dec 21 2008
So T(n,k) = Stirling1(n,k) + n*Stirling1(n-1,k), Stirling1 being the (signed) Stirling numbers of first kind A048994. In terms of lower triangular matrices, 0<= k <= n, T is also the product [Stirling1] * [Pascal] = [A048994] * [A007318], i.e., T(n,k) = Sum_{j=0..n} Stirling1(n,j) * binomial(j,k). - Giuliano Cabrele, Jan 19 2009
This is the triangle of connection constants for expressing the basis of falling factorial polynomials x_(k) := x*(x-1)*...*(x-k+1) in terms of the polynomial sequence (x-1)^n, that is, x_(n) = Sum_{k = 0..n} T(n,k)*(x-1)^k. - Peter Bala, Jul 10 2013
From Wolfdieter Lang, Jun 19 2017: (Start)
Triangle T is the (infinite) matrix product of A048994 (Stirling1) and A007318 (Pascal): T(n,k) = Sum_{m=k..n} Stirling1(n, m)*Pascal(m, k), n >= k >= 0, and 0 for n < k. Note that the Pascal matrix is Sheffer (e^t, t) of the Appell type.
T is the Sheffer (aka exponential Riordan) matrix (1+t, log(1+t)).
E.g.f. column k: (1+x)*(log(1+x))^k/k!, k >= 0.
The a-sequence for T is A027641/A027642 (Bernoulli), and the z-sequence is A033999 (repeat(1,-1)) (see a W. Lang link under A006232 for a- and z-sequences for Sheffer matrices, also for references).
Therefore the combined recurrence is: T(n, 0) = n*Sum_{j=0..n-1} (-1)^j*T(n-1, j), n >= 1, T(0, 0) = 1, and T(n, m) = (n/m)*Sum_{j=0..n-m} binomial(m-1+j, m-1)*Bernoulli(j)*T(n-1, m-1+j), n >= m >= 1. (End)

A269954 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 5, 3, 1, 0, 9, 20, 17, 6, 1, 0, 44, 109, 100, 45, 10, 1, 0, 265, 689, 694, 355, 100, 15, 1, 0, 1854, 5053, 5453, 3094, 1015, 196, 21, 1, 0, 14833, 42048, 48082, 29596, 10899, 2492, 350, 28, 1

Offset: 0

Peter Luschny, Apr 12 2016

			Triangle starts:
  1;
  0,   1;
  0,   0,   1;
  0,   1,   1,   1;
  0,   2,   5,   3,   1;
  0,   9,  20,  17,   6,   1;
  0,  44, 109, 100,  45,  10,  1;
  0, 265, 689, 694, 355, 100, 15, 1;

Peter Luschny, Extensions of the binomial

A000255 (row sums), A000217 (diag. n,n-1), A133252 (diag. n,n-2).
Columns k=0..4 give A000007, A000166(n-1), A300490(n-1), A381067(n-1), A381068(n-1).
Cf. A269951, A269953.

A269954 := (n, k) -> add(binomial(-j, -n)*abs(Stirling1(j, k)), j=0..n):
seq(seq(A269954(n, k), k=0..n), n=0..9);

Flatten[Table[Sum[Binomial[-j,-n] Abs[StirlingS1[j,k]],{j,0,n}], {n,0,9},{k,0,n}]]

T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(n-1, n-j)*abs(stirling(j, k)));
for(n=0, 9, for(k=0, n, print1(T(n, k), ", "))); \\ Seiichi Manyama, Feb 13 2025

A381065 Expansion of e.g.f. -log(1-x)^3 * exp(-x) / 6.

Original entry on oeis.org

0, 0, 0, 1, 2, 15, 85, 609, 4844, 43238, 427090, 4630241, 54683046, 699012093, 9617979007, 141755256889, 2228396376088, 37221746535564, 658390407698084, 12295201090394017, 241749652842156074, 4992277083472634507, 108032799218176059337, 2444797394606939402449

Offset: 0

Seiichi Manyama, Feb 12 2025

nonn

Column k=3 of A269953.

Cf. A381022, A381067.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(stirling(k, 3, 1)));

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * |Stirling1(k,3)|.

A381066 Expansion of e.g.f. log(1-x)^4 * exp(-x) / 24.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 40, 315, 2779, 26817, 282785, 3240325, 40144126, 535152332, 7642713715, 116465389950, 1886911421914, 32395513943998, 587627463812070, 11231176543495238, 225621300685737631, 4753177896741075823, 104793882332694641218, 2413274241933067193021

Offset: 0

Seiichi Manyama, Feb 12 2025

nonn

Column k=4 of A269953.

Cf. A381023, A381068.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(stirling(k, 4, 1)));

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * |Stirling1(k,4)|.

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.

Original entry on oeis.org

1, -2, 1, 4, -3, 1, -8, 8, -3, 1, 16, -18, 11, -2, 1, -32, 44, -20, 15, 0, 1, 64, -80, 94, 5, 25, 3, 1, -128, 272, 56, 294, 105, 49, 7, 1, 256, 112, 1868, 1596, 1169, 392, 98, 12, 1, -512, 5280, 12216, 16148, 10290, 4305, 1092, 186, 18, 1

Offset: 0

Igor Victorovich Statsenko, Feb 13 2025

			Triangle starts:
  [0]     1;
  [1]    -2,      1;
  [2]     4,     -3,       1;
  [3]    -8,      8,      -3,       1;
  [4]    16,    -18,      11,      -2,       1;
  [5]   -32,     44,     -20,      15,       0,        1;
  [6]    64,    -80,      94,       5,      25,        3,     1;
  [7]  -128,    272,      56,     294,     105,       49,     7,     1;
  [8]   256,    112,    1868,    1596,    1169,      392,    98,    12,    1;
  [9]  -512,   5280,   12216,   16148,   10290,     4305,  1092,   186,   18,     1;
  ...

Cf. A000023 (row sums).
Columns 0,1: A122803, A346397.
Triangles: for m = -3 is A327997; for m = -2 is A137346 (unsigned); for m = -1 is A094816; for m = 0 is A132393; for m = 1 is A269953.

T:=(m,n,k)->add(Stirling1(n-i,k)*binomial(n,i)*m^(i)*(-1)^(n-k), i=0..n):
m:=2:seq(print(seq(T(m,n,k), k=0..n)), n=0..9);

T(n,k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), where m = 2.

A381064 Expansion of e.g.f. log(1-x)^2 * exp(-x) / 2.

Original entry on oeis.org

0, 0, 1, 0, 5, 15, 94, 595, 4458, 37590, 354051, 3682646, 41935695, 518954293, 6935360496, 99553094537, 1527716784020, 24959724735564, 432572721886437, 7926615468800172, 153129657663788761, 3110514839038091643, 66278515188844197218, 1478222957082474301887

Offset: 0

Seiichi Manyama, Feb 12 2025

nonn

Column k=2 of A269953.

Cf. A300490, A381021.

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*abs(stirling(k, 2, 1)));

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * |Stirling1(k,2)|.

A002633 Related to discordant permutations.

Original entry on oeis.org

1, -3, 5, -3, 9, -3, -51, -675, -5871, -46467, -331371, -1852227, -920295, 224455293, 5571057501, 104877816093, 1781775072801, 28519837563645, 431525731169061, 5994769814117757, 68879336771960361, 346333945918252797, -15047168730918615315, -793523760950138583843

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

K. Yamamoto, Structure polynomial of Latin rectangles and its application to a combinatorial problem, Memoirs of the Faculty of Science, Kyusyu University, Series A, 10 (1956), 1-13.
K. Yamamoto, Structure polynomial of Latin rectangles and its application to a combinatorial problem, Memoirs of the Faculty of Science, Kyusyu University, Series A, 10 (1956), 1-13. [Annotated scanned copy]

Cf. A213170, A269953.

a[ n_ ] := a[ n ]=(2n-5)a[ n-1 ]-(n-1)(n-4)a[ n-2 ]-(n-1)(n-2)a[ n-3 ]; a[ 0 ]=1; a[ 1 ]=-3; a[ 2 ]=5; Table[ a[ n ], {n, 0, 24} ] (* Typo fixed by Vaclav Kotesovec, Mar 20 2014 *)

a(n) - (2n-5)*a(n-1) + (n-1)*(n-4)*a(n-2) + (n-1)*(n-2)*a(n-3) = 0.
From Mélika Tebni, Mar 02 2022: (Start)
a(n) = Sum_{k=0..n} A213170(k)*A269953(n, k).
E.g.f.: exp(-x*(3 - x) / (1 - x)). (End)

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A001113 Decimal expansion of e.

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A105793 Expansion of e.g.f. (1 + y)^(1 + x).

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A269954 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

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A381065 Expansion of e.g.f. -log(1-x)^3 * exp(-x) / 6.

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A381066 Expansion of e.g.f. log(1-x)^4 * exp(-x) / 24.

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A381082 Triangle T(n,k) read by rows, where the 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=2.

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Maple

Formula

A381064 Expansion of e.g.f. log(1-x)^2 * exp(-x) / 2.

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PARI

Formula

A002633 Related to discordant permutations.

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Mathematica

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Extensions

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.