cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A155856 Triangle T(n,k) = binomial(2*n-k, k)*(n-k)!, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 6, 1, 24, 42, 30, 10, 1, 120, 216, 168, 70, 15, 1, 720, 1320, 1080, 504, 140, 21, 1, 5040, 9360, 7920, 3960, 1260, 252, 28, 1, 40320, 75600, 65520, 34320, 11880, 2772, 420, 36, 1, 362880, 685440, 604800, 327600, 120120, 30888, 5544, 660, 45, 1
Offset: 0

Views

Author

Paul Barry, Jan 29 2009

Keywords

Comments

Row sums of B^{-1}*A155856*B^{-1} are A000166 with B=A007318.
Downward diagonals T(n+j, n) = j!*binomial(n+j, n) = j!*seq(j), where seq(j) are sequences A010965, A010967, ..., A011101, A017714, A017716, ..., A017764, for 6 <= j <= 50, respectively. - G. C. Greubel, Jun 04 2021

Examples

			Triangle begins:
     1;
     1,    1;
     2,    3,    1;
     6,   10,    6,    1;
    24,   42,   30,   10,    1;
   120,  216,  168,   70,   15,   1;
   720, 1320, 1080,  504,  140,  21,  1;
  5040, 9360, 7920, 3960, 1260, 252, 28, 1;

Links

Crossrefs

Cf. A155857 (row sums), A155858 (diagonal sums).

Programs

  • Mathematica
    Table[Binomial[2n-k,k](n-k)!,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 24 2017 *)
  • Sage
    flatten([[factorial(n-k)*binomial(2*n-k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 04 2021

Formula

T(n,k) = binomial(2*n-k, k)*(n-k)!.
Sum_{k=0..n} T(n, k) = A155857(n)
Sum_{k=0..floor(n/2)} T(n-k, k) = A155858(n) (diagonal sums).
G.f.: 1/(1-xy-x/(1-xy-x/(1-xy-2x/(1-xy-2x/(1-xy-3x/(1-.... (continued fraction).
From G. C. Greubel, Jun 04 2021: (Start)
T(n, 0) = A000142(n). T(n+1, n) = A000217(n+1).
T(n+1, 1) = A007680(n). T(n+2, n) = A034827(n+4).
T(n+2, 2) = A175925(n). T(n+3, n) = A253946(n).
T(2*n, n) = A064352(n) T(n+4, n) = 4!*A000581(n).
T(n+1, n) = A000217(n+1). T(n+5, n) = 5!*A001287(n). (End)

A018159 Powers of fifth root of 16 rounded down.

Original entry on oeis.org

1, 1, 3, 5, 9, 16, 27, 48, 84, 147, 256, 445, 776, 1351, 2352, 4096, 7131, 12416, 21618, 37640, 65536, 114104, 198668, 345901, 602248, 1048576, 1825676, 3178688, 5534417, 9635980, 16777216, 29210829, 50859008, 88550676, 154175683
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A011101.

Programs

  • Magma
    seq := [Floor(16^(n/5)) : n in [0..40]]; seq; // Vincenzo Librandi, Jun 06 2025
  • Mathematica
    Floor[(Power[16, (5)^-1])^Range[0,40]]  (* Harvey P. Dale, Jan 22 2011 *)

A018160 Powers of fifth root of 16 rounded to nearest integer.

Original entry on oeis.org

1, 2, 3, 5, 9, 16, 28, 49, 84, 147, 256, 446, 776, 1351, 2353, 4096, 7132, 12417, 21619, 37641, 65536, 114105, 198668, 345901, 602249, 1048576, 1825677, 3178688, 5534417, 9635980, 16777216, 29210830, 50859008
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A011101.

Programs

  • Mathematica
    Floor[(16^(1/5))^Range[0,40]+1/2] (* Harvey P. Dale, May 23 2012 *)
  • Python
    from gmpy2 import iroot_rem
def A018160(n):
    i, j = iroot_rem(1<<(n<<2),5)
    return int(i)+int(j<<5>=10*i*((i*((i*(i+1)<<1)+1)<<2)+1)+1) # Chai Wah Wu, Jun 20 2024

A281143 Decimal expansion of 10!^(1/10).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Jan 15 2017

Keywords

Comments

Base b such that log_b 10! = 10.
Inspired by the idea of utilizing the log scaled to 10! being 10, i.e., log_b 10! = 10, therefore b = 2^(4/5)*3^(2/5)*5^(1/5)*7^(1/10).

Examples

			4.52872868811676476220330933719550879349863167608939046288611476046926255...

Crossrefs

Cf. A002193, A005486, A005534, A011020, A011092, A011094, A011101.

Programs

  • Mathematica
    RealDigits[2^(4/5) 3^(2/5) 5^(1/5) 7^(1/10), 10, 111][[1]] (* or *)
RealDigits[Solve[Log[b, 10!] == 10, b][[1, 1, 2]], 10, 105][[1]]

Formula

Equals A011101 * A011094 * A005534 * A011092.

A358938 Decimal expansion of the real root of 2*x^5 - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Dec 07 2022

Keywords

Comments

This is the reciprocal of A005531.
The other two complex conjugate pairs of roots are obtained, with the present number r = (1/2)^(1/5) and the golden section phi (A001622), from x1 = r*exp(Pi*i*2/5) = r*(phi - 1 + sqrt(2 + phi)*i)/2 = r*(A001622 - 1 + A188593*i)/2 = 0.2690149185... + 0.8279427859...*i, x2 = r*exp(Pi*i*4/5) = r*(-phi + sqrt(3 - phi)*i)/2 = r*(-A001622 + A182007*i)/2 = -0.7042902001... + 0.5116967824...*i.

Examples

			0.87055056329612413913627001747974609897912542434800304824185956850675...

Crossrefs

Cf. A001622, A182007, A188593, A005531, A011101.

Programs

  • Mathematica
    RealDigits[Surd[1/2, 5], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)

Formula

r = (1/2)^(1/5) = 1/A005531.
Equals A011101/2. - Hugo Pfoertner, Mar 24 2025
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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