A095884 -id:A095884 - 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

A009999 Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

T(n,k) is the number of ways of placing 1..k in n boxes such that each box contains at most one number, and numbers in adjacent boxes are in increasing order. This can be proved by observing that there are n-(k-1) ways of extending each of T(n-1,k-1). - Jimin Park, Apr 16 2023

The n-th diagonal consists of n^k. This can also be generated as the Akiyama-Tanigawa algorithm applied to the sequence binomial(n+k,k). - Shel Kaphan, May 03 2024

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 24.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Row sums give A026898.
T(2n,n) gives A000169(n+1).
Cf. A095884, A051128.
Cf. A009998 (mirrored).

a009999 n k = (n + 1 - k) ^ k
a009999_row n = a009999_tabl !! n
a009999_tabl = [1] : map snd (iterate f ([1,1], [1,1])) where
   f (us@(u:_), vs) = (us', 1 : zipWith (*) us' vs)
                      where us' = (u + 1) : us
-- Reinhard Zumkeller, Feb 02 2014

A009999 := proc(i,j) (i+1-j)^j ; end proc: # R. J. Mathar, Jan 16 2011

Table[(i+1-j)^j, {i, 0, 10}, {j, 0, i}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

T(n,0) = 1; T(n,k) = (n-k+1)*T(n-1,k-1) for k=1..n. - Reinhard Zumkeller, Feb 02 2014

T(n,k) = Sum_{i=0..k} binomial(k,i)*T(n-1-i,k-i). - Jimin Park, Apr 16 2023

T(10,8) corrected by Reinhard Zumkeller, Feb 02 2014

A095885 Let F(x) be the function such that F(F(x)) = arctan(x), then F(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.

Original entry on oeis.org

1, -1, 7, -129, 4489, -249485, 20130311, -2233371917, 329796121169, -62717244921977, 14635852695795623, -4021254983530551833, 1345410098638648813593, -602107783524378517901957, 304930961594040255078182951, -81321478646591075063015099621

Offset: 0

Paul D. Hanna, Jun 11 2004

sign

			F(x) = x - 1*x^3/3! + 7*x^5/5! - 129*x^7/7! + 4489*x^9/9! - 249485*x^11/11! +- ...
Special values:
F(x)=Pi/8 at x=F(sqrt(2)-1) = 0.40303074376779286039477674...
F(x)=Pi/6 at x=F(sqrt(3)/3) = 0.54854553000356...

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A095882, A095883, A095884, A048605.

T(n,m):=if n=m then 1 else 1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum((2^i*stirling1(i,m)*binomial(n-1,i-1))/i!,i,m,n)-sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist((2*n-1)!*T(2*n-1,1),n,1,5); /* Vladimir Kruchinin, Mar 12 2012 */

{a(n)=local(A,B,F);F=atan(x+x*O(x^(2*n+1)));A=F; for(i=0,n,B=serreverse(A);A=(A+subst(B,x,F))/2);(2*n+1)!*polcoeff(A,2*n+1,x)}

a(n) = (2*n-1)!*T(2*n-1,1), T(n,m)=1/2*(2^(-m-1)*m!*((-1)^(n+m)+1)*(-1)^((3*n+m)/2)*sum(i=m..n, (2^i*stirling1(i,m)*binomial(n-1,i-1))/i!)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 12 2012

A095883 Let F(x) be the function such that F(F(x)) = arcsin(x), then F(x) = Sum_{n>=0} a(n)/2^n*x^(2n+1)/(2n+1)!.

Original entry on oeis.org

1, 1, 13, 501, 38617, 4945385, 944469221, 250727790173, 88106527550129, 39555449833828817, 22093952731139969213, 15041143328788464370373, 12273562321018687866908553, 11833097802606125967312406457

Offset: 0

Paul D. Hanna, Jun 10 2004

It appears that, if arcsin(x) is changed to arcsinh(x) in the definition, the sequence obtained is the same except alternating in sign: 1, -1, 13, -501, ... - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Jul 16 2009

a(35) is negative. - Vaclav Kotesovec, Jan 06 2023

			F(x) = x + (1/2)*x^3/3! + (13/2^2)*x^5/5! + (501/2^3)*x^7/7! + (38617/2^4)*x^9/9! + ...
Special values:
F(x)=Pi/6 at x=F(1/2) = 0.51137532057552418592144885355...
F(x)=Pi/4 at x=F(sqrt(2)/2) = 0.74287348600976...

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A095882, A095884, A095885.

a[n_] := Module[{A, B, F}, F = ArcSin[x] + O[x]^(2n+3); A = F; For[i = 0, i <= n, i++, B[x_] = InverseSeries[A, x] // Normal; A = (A + B[F])/2]; 2^n* (2n+1)!*SeriesCoefficient[A, {x, 0, 2n+1}]];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Aug 16 2022, after PARI code *)

{a(n)=local(A,B,F);F=asin(x+x*O(x^(2*n+1)));A=F; for(i=0,n,B=serreverse(A);A=(A+subst(B,x,F))/2);2^n*(2*n+1)!*polcoeff(A,2 *n+1,x)}

a(n)=T(2*n+1,1)*2^n*(2*n+1)!, T(n,m)=if n=m then 1 else 1/2(Co(n,m)-sum(i=m+1..n-1, T(n,i)*T(i,m))), Co(n,m)=T121408(n,m)=(m!*(sum(k=0..n-m, (-1)^((k)/2)*(sum(i=0..k, (2^i*stirling1(m+i,m)* binomial(m+k-1,m+i-1))/(m+i)!))*binomial((n-2)/2,(n-m-k)/2)))*((-1)^(n-m)+1))/2. - Vladimir Kruchinin, Nov 11 2011

A329940 Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.

Original entry on oeis.org

1, 3, 2, 7, 8, 3, 15, 26, 15, 4, 31, 80, 63, 24, 5, 63, 242, 255, 124, 35, 6, 127, 728, 1023, 624, 215, 48, 7, 255, 2186, 4095, 3124, 1295, 342, 63, 8, 511, 6560, 16383, 15624, 7775, 2400, 511, 80, 9, 1023, 19682, 65535, 78124, 46655, 16806, 4095, 728, 99, 10

Offset: 1

Roy S. Freedman, Nov 24 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B. A relation R is right unique if (a, b1) in R and (a,b2) in R implies b1=b2. T(n,k) is the number of right unique relations and T(k,n) is the number of left unique relations: relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.

			T(n,k) begins:
    1,    2,     3,      4,       5,       6,        7,        8, ...
    3,    8,    15,     24,      35,      48,       63,       80, ...
    7,   26,    63,    124,     215,     342,      511,      728, ...
   15,   80,   255,    624,    1295,    2400,     4095,     6560, ...
   31,  242,  1023,   3124,    7775,   16806,    32767,    59048, ...
   63,  728,  4095,  15624,   46655,  117648,   262143,   531440, ...
  127, 2186, 16383,  78124,  279935,  823542,  2097151,  4782968, ...
  255, 6560, 65535, 390624, 1679615, 5764800, 16777215, 43046720, ...

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

Cf. A037205 (main diagonal).

Cf. A003992, A004248, A009998, A009999, A051128, A051129, A095884.

T:= (n, k)-> (k+1)^n-1:
seq(seq(T(1+d-k, k), k=1..d), d=1..12);

T[n_, k_] := (k + 1)^n - 1; Table[T[n - k + 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

MuPAD
```
T:=(n,k)->(k+1)^n-1:
```

T(n,k) = (k+1)^n - 1.

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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A009999 Triangle in which j-th entry in i-th row is (i+1-j)^j, 0<=j<=i.

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A095885 Let F(x) be the function such that F(F(x)) = arctan(x), then F(x) = Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)!.

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A095883 Let F(x) be the function such that F(F(x)) = arcsin(x), then F(x) = Sum_{n>=0} a(n)/2^n*x^(2n+1)/(2n+1)!.

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A329940 Square array read by antidiagonals upwards: T(n,k) is the number of right unique relations between set A with n elements and set B with k elements.

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