A010842 -id:A010842 - cp's OEIS frontend

A300484 a(n) = 2 * Integral_{t>=0} T_n(t/2+1) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

2, 3, 8, 29, 130, 697, 4376, 31607, 258690, 2368847, 24011832, 267025409, 3233119106, 42346123861, 596617706344, 8998126507307, 144651872924162, 2469279716419035, 44609768252582312, 850345380011532261, 17056474009400181122

Offset: 0

Max Alekseyev, Mar 06 2018

nonn

For any integer n>=0, 2 * Integral_{t=-2..2} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-1..1} T_n(z)*exp(-2*z)*dz = A102761(n)*exp(2) - a(n)*exp(-2).

Row m=2 in A300480.
Row sums of A156995.
Cf. A102761, A300482, A300483, A300485.

{ A300484(n) = if(n==0, return(2)); subst( serlaplace( 2*polchebyshev(n, 1, (x+2)/2)), x, 1); }

a(n) = Sum_{i=0..n} A127672(n,i) * A010842(i).
a(n) = A300480(2,n) = A300481(-2,n).
a(n) = Sum_{m=0..n} A156995(n,m) = 2*n*Sum_{m=0..n} binomial(2*n-m, m)*(n-m)!/(2*n-m).

A080955 Square array of numbers related to the incomplete gamma function, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 6, 1, 4, 10, 16, 24, 1, 5, 17, 38, 65, 120, 1, 6, 26, 78, 168, 326, 720, 1, 7, 37, 142, 393, 872, 1957, 5040, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1, 10, 82, 538, 2760, 10970, 34960, 100026

Offset: 0

Paul Barry, Feb 26 2003

			Array begins:
k=0: 1 1 2 6 24 ...
k=1: 1 2 5 16 65 ...
k=2: 1 3 10 38 168 ...
k=3: 1 4 17 78 393 ...
k=4: 1 5 26 142 824 ...
...

Rows: A000142, A000522, A010842, A053486, A053487, A080954.

Transposed version: A089258.

T[0, k_] := k!; T[n_, k_] := k!*Sum[n^j/j!, {j, 0, k}];
Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 17 2018 *)

T(k,n) = n! * Sum{j=0..n} k^j/j!.
E.g.f. of k-th row: exp(k*x)/(1-x).
T(k,n) = A089258(n,k).

Corrected by Philippe Deléham, Dec 12 2003

A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 9, 1, 4, 10, 16, 24, 44, 1, 5, 17, 38, 65, 120, 265, 1, 6, 26, 78, 168, 326, 720, 1854, 1, 7, 37, 142, 393, 872, 1957, 5040, 14833, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 133496, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1334961

Offset: 0

Philippe Deléham, Dec 12 2003

Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018

			n\k -1   0   1    2    3    4     5     6  ...
----------------------------------------------
0  | 1,  1,  1,   1,   1,   1,    1,    1, ...
1  | 0,  1,  2,   3,   4,   5,    6,    7, ...
2  | 1,  2,  5,  10,  17,  26,   37,   50, ...
3  | 2,  6, 16,  38,  78, 152,  236,  366, ...
4  | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
...

Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.
Main diagonal gives A217701.
Cf. A080955, A008290, A292977.

(* Assuming offset (0, 0): *)
T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, Dec 24 2021 *)

For n > 0, k >= -1, T(n,k) is the permanent of the n X n matrix with k+1 on the diagonal and 1 elsewhere.
T(0,k) = 1.
T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.
T(n,k) = n*T(n-1, k) + k^n .
T(n,k) = n! * Sum_{j=0..n} k^j/j!.
E.g.f. for k-th column: exp(k*x)/(1-x).
Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - Peter Luschny, Dec 24 2021

Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018

A093658 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 6, 2, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 6, 2, 2, 1, 0, 0, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 6, 2, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1

Offset: 1

Paul D. Hanna, Apr 08 2004

Related to factorials, the incomplete gamma function (A010842) and the total number of arrangements of sets (A000522).
First column forms A093659, where A093659(2^n) = n! for n>=0.
Row sums form A093660, where A093660(2^n) = A000522(n) for n>=0.
Partial sums of the row sums form A093661, where A093661(2^n) = A010842(n) for n>=0.

			Let M(n) be the lower triangular matrix formed from the first 2^n rows.
To generate M(3) from M(2), take the matrix square of M(2):
[1,0,0,0]^2=[1,0,0,0]
[1,1,0,0]...[2,1,0,0]
[1,0,1,0]...[2,0,1,0]
[2,1,1,1]...[6,2,2,1]
and append M(2)^2 to the bottom left corner and M(2) to the bottom right:
[1],
[1,1],
[1,0,1],
[2,1,1,1],
.........
[1,0,0,0],[1],
[2,1,0,0],[1,1],
[2,0,1,0],[1,0,1],
[6,2,2,1],[2,1,1,1].
Repeating this process converges to triangle A093658.

Cf. A000522, A010842, A093655, A093662.

T(2^n, 1) = n! for n>=0.

A081923 Expansion of e.g.f.: exp(2x)/(1-x)^2.

Original entry on oeis.org

1, 4, 18, 92, 536, 3552, 26608, 223456, 2085504, 21450752, 241320704, 2949474816, 38933066752, 552141672448, 8374148696064, 135274709700608, 2318995023429632, 42051109758173184, 804227474125029376

Offset: 0

Paul Barry, Apr 01 2003

Binomial transform of A001339.
Polynomials in A010027 evaluated at 3. - Ralf Stephan, Dec 15 2004
From Dennis P. Walsh, Sep 18 2013: (Start)
a(n) is the number of rooted labeled forests that satisfy the following conditions:
(i) there are 4 roots labeled 1, 2, 3, and 4;
(ii) there are n non-root vertices labeled 5,..., n+4;
(iii) the trees with roots 1 and 2 have width one;
(iv) the trees with roots 3 and 4 have height at most one.
To construct such a forest, for k=0,...,n, we take the following steps:
(1) choose k non-root vertices for trees with roots 1 and 2;
(2) construct width-one trees on roots 1 and 2 with the k non-root vertices;
(3) with the n-k remaining non-root vertices construct trees of height at most one on roots 3 and 4.
Thus a(n) is the sum (over k) of the product of the number of ways to do each step: a(n)=sum(k=0..n, binomial(n,k)*(k+1)!*2^(n-k)).  (End)

			For n=2, the a(2)=18 forests that satisfy the specified conditions are given in the link above. - _Dennis P. Walsh_, Sep 20 2013

Harvey P. Dale, Table of n, a(n) for n = 0..448
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
Dennis P. Walsh, 18 Forests

Cf. A081924. A010842.

Cf. A081923(n) (sum(k=0..n, binomial(n,k)*A000522(n-k)*A000522(k))).

seq(n!*add((k+1)*2^(n-k)/(n-k)!,k=0..n),n=0..40); # Dennis P. Walsh, Sep 18 2013
seq(simplify(KummerU(-n, -n - 1, 2)), n = 0..24); # Peter Luschny, May 10 2022

With[{nn=20},CoefficientList[Series[Exp[2x]/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 10 2025 *)

E.g.f.: exp(2*x)/(1-x)^2
E.g.f.: 1/U(0) where U(k)=   1 - 2*x/( 1 + x/(2 - x - 4/( 2 - x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
Conjecture: a(n) +(-n-3)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
G.f.: 2/x/G(0) - 1/x, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+4) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: (sum(k>=0, k!*(x/(1-2*x))^k ) - 1)/x = Q(0)/(2*x) - 1/x, where Q(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-2*x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(k+1)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) = n!*sum(k=0..n, (k+1)*2^(n-k)/(n-k)!). [Dennis P. Walsh, Sep 18 2013]
a(n) = n!*sum(k=0..n, (n-k+1)*2^k/k!). [Dennis P. Walsh, Sep 18 2013]
From Peter Bala, Sep 25 2013: (Start)
a(n) ~ n!*n*e^2.
Applying Maple's ZeilbergerRecurrence command to the above series of Walsh for a(n) results in the first-order recurrence equation (n - 1)*a(n+1) = n*(n + 1)*a(n) - 2^(n+2) with a(0) = 1 and a(2) = 18. Using this it is easy to verify that a(n) satisfies the second-order recurrence a(n) = (n + 3)*a(n-1) - 2*(n - 1)*a(n-2) conjectured above by Mathar.
The sequence b(n) = n!*(n - 1) satisfies the same second-order recurrence but with the initial conditions b(0) = -1 and b(1) = 0. This leads to the finite continued fraction expansion a(n)/b(n) = 9 - 2*( 4/(6 - 6/(7 - 8/(9 - ... - 2*n/(n + 4)))) ) valid for n >= 2. Letting n tend to infinity produces the infinite continued fraction expansion e^2 = 9 - 2*( 4/(6 - 6/(7 - 8/(9 - ... - 2*n/(n + 4 - ...)))) ). (End)
a(n) = KummerU(-n, -n - 1, 2). - Peter Luschny, May 10 2022

Definition clarified by Harvey P. Dale, May 10 2025

A346394 Expansion of e.g.f. -log(1 - x) * exp(2*x).

Original entry on oeis.org

0, 1, 5, 20, 78, 324, 1520, 8336, 53872, 405600, 3492416, 33798016, 362543104, 4264455168, 54540715008, 753246711808, 11168972683264, 176937613586432, 2982069587042304, 53271637651996672, 1005385746384846848, 19987620914387812352, 417489079682758213632

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A002104, A010842, A066534, A346395, A346396.

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[2^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+1)*v[i]-2*(i-1)*v[i-1]+2^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} 2^k / ((n-k) * k!).
a(n) = Sum_{k=0..n} binomial(n,k) * A002104(k).
a(n) ~ exp(2) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+1) * a(n-1) - 2 * (n-1) * a(n-2) + 2^(n-1). - Seiichi Manyama, May 27 2022

A053484 Numerators in expansion of exp(2x)/(1-x).

Original entry on oeis.org

1, 3, 5, 19, 7, 109, 331, 155, 2327, 20947, 34913, 164591, 691283, 14977801, 314533829, 4718007451, 1572669151, 16041225341, 103122162907, 4571749222213, 68576238333199, 110777000384399, 55582845806909, 364345554264288511

Offset: 0

N. J. A. Sloane, Jan 15 2000

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A053485, A010842.

Numerator[Table[Sum[2^k/k!,{k,0,n}],{n,0,30}]] (* Alexander Adamchuk, Jul 22 2006 *)
Numerator[Accumulate[Table[2^n/n!,{n,0,30}]]] (* or *) Numerator[ CoefficientList[ Series[Exp[2x]/(1-x),{x,0,30}],x] ] (* Harvey P. Dale, Mar 12 2017 *)

for(n=0,25, print1(numerator(sum(k=0,n, 2^k/k!)), ", ")) \\ G. C. Greubel, Jan 18 2017

a(n) = Numerator[Sum[2^k/k!,{k,0,n}]]. - Alexander Adamchuk, Jul 22 2006

A053485 Denominators in expansion of exp(2x)/(1-x).

Original entry on oeis.org

1, 1, 1, 3, 1, 15, 45, 21, 315, 2835, 4725, 22275, 93555, 2027025, 42567525, 638512875, 212837625, 2170943775, 13956067125, 618718975875, 9280784638125, 14992036723125, 7522320180375, 49308808782358125, 147926426347074375

Offset: 0

N. J. A. Sloane, Jan 15 2000

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A053484, A010842.

Denominator[Table[Sum[2^k/k!, {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Jan 18 2017 *)

for(n=0,25, print1(denominator(sum(k=0,n, 2^k/k!)), ", ")) \\ G. C. Greubel, Jan 18 2017

a(n) = denominator( Sum_{k=0..n} 2^k/k! ). - G. C. Greubel, Jan 18 2017

A331689 E.g.f.: exp(x/(1 - x)) / (1 - 2*x).

Original entry on oeis.org

1, 3, 15, 103, 897, 9471, 117703, 1685475, 27361953, 497111707, 10001175231, 220849928223, 5312868439585, 138337555830423, 3876986580776247, 116375171226474331, 3725295913465848513, 126686907674290095795, 4561317309742758852463, 173343622143918424951767

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..403

Cf. A000165, A000262, A000522, A002720, A010842, A010844, A331688.

b:= proc(n) b(n):= `if`(n<0, 0, 1+n*b(n-1)) end:
a:= n-> n!*add(binomial(n, k)*b(k)/k!, k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Jan 24 2020

nmax = 19; CoefficientList[Series[Exp[x/(1 - x)]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
A000522[0] = 1; A000522[n_] := Floor[Exp[1] n!]; a[n_] := Sum[Binomial[n, k]^2 k! A000522[n - k], {k, 0, n}]; Table[a[n], {n, 0, 19}]

a(n) = Sum_{k=0..n} binomial(n,k)^2 * k! * A000522(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * k! * 2^k * A000262(n-k).
a(n) ~ n! * exp(1) * 2^n. - Vaclav Kotesovec, Jan 26 2020

A093660 Row sums of lower triangular matrix A093658.

Original entry on oeis.org

1, 2, 2, 5, 2, 5, 5, 16, 2, 5, 5, 16, 5, 16, 16, 65, 2, 5, 5, 16, 5, 16, 16, 65, 5, 16, 16, 65, 16, 65, 65, 326, 2, 5, 5, 16, 5, 16, 16, 65, 5, 16, 16, 65, 16, 65, 65, 326, 5, 16, 16, 65, 16, 65, 65, 326, 16, 65, 65, 326, 65, 326, 326, 1957

Offset: 1

Paul D. Hanna, Apr 08 2004

nonn

Records form A000522 (total number of arrangements of set of n elements). Partial sums form A093661, where A093661(2^n) = A010842(n) (incomplete Gamma Function at 2).

Cf. A000522, A010842, A093658, A093661.

a(2^n) = A000522(n) for n>=0. a(2^n+2^m) = a(2^(m+1)) for n>m>=0.

cp's OEIS Frontend

A300484 a(n) = 2 * Integral_{t>=0} T_n(t/2+1) * exp(-t) * dt, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

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A080955 Square array of numbers related to the incomplete gamma function, read by antidiagonals.

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A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

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A093658 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)]], with M(0) = [1].

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A081923 Expansion of e.g.f.: exp(2x)/(1-x)^2.

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Maple

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A346394 Expansion of e.g.f. -log(1 - x) * exp(2*x).

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A053484 Numerators in expansion of exp(2x)/(1-x).

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A053485 Denominators in expansion of exp(2x)/(1-x).

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