A099555 -id:A099555 - cp's OEIS frontend

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A099554 Decimal expansion of the constant x that satisfies x = exp(1/sqrt(x)).

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Oct 22 2004

This constant arises from the series: S(n) = Sum_{k=0..2n} (n-[k/2])^k/k!. The asymptotic behavior of this series is given by: S(n) ~ c*x^n where c = (x+sqrt(x))/(1+2*sqrt(x)) = 0.8957126... and x = 2.0207473586... satisfies x = exp(1/sqrt(x)).

			x=2.02074735861185766811269528724732366499433113141625298973171160826928577...
To demonstrate how this constant describes the asymptotics of the sum:
S(n) = Sum_{k=0..2n} (n-[k/2])^k/k! ~ c*x^n
evaluate the sum at n=5:
S(5) = 1+ 5+ 4^2/2!+ 4^3/3!+ 3^4/4!+ 3^5/5!+ 2^6/6!+ 2^7/7!+ 1/8!+ 1/9!
= 782291/25920 = 30.1809799... = (0.89572199...)*x^5
and evaluate the sum at n=6:
S(6) = 1+ 6+ 5^2/2!+ 5^3/3!+ 4^4/4!+ 4^5/5!+ 3^6/6!+ 3^7/7!+ 2^8/8!+ 2^9/9!+ 1/10!+ 1/11!
= 608606683/9979200 = 60.9875223... = (0.89571298...)*x^6.

Cf. A099555, A202356.

RealDigits[x/.FindRoot[x==Exp[1/Sqrt[x]],{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 06 2013 *)
RealDigits[ 1/(4*ProductLog[1/2]^2), 10, 105] // First (* Jean-François Alcover, Feb 15 2013 *)

PARI
```
solve(x=2,2.1,x-exp(1/sqrt(x)))
```

Equals 1/(4*A202356^2). - Vaclav Kotesovec, Oct 06 2020

A099556 a(n) = Sum_{k=0..2*n} (n - floor(k/2))^k.

Original entry on oeis.org

1, 2, 5, 18, 91, 604, 5123, 53808, 679389, 10107934, 174544505, 3451802006, 77302457079, 1942469215432, 54339334903015, 1680597906265988, 57116435383949097, 2121725700531106842, 85740960774906267853, 3753239245360442525498, 177284859933198522968819, 9004760569281510790598100

Offset: 1

Paul D. Hanna, Oct 22 2004

Cf. A099555.

PARI
```
a(n)=sum(k=0,2*n,(n-k\2)^k)
```

Row sums of triangle A099555.

a(13) and following terms from Georg Fischer, Nov 21 2024

cp's OEIS Frontend

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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A099554 Decimal expansion of the constant x that satisfies x = exp(1/sqrt(x)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A099556 a(n) = Sum_{k=0..2*n} (n - floor(k/2))^k.

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PARI

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Extensions