A081567 -id:A081567 - cp's OEIS frontend

A081572 Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 13, 5, 1, 5, 17, 35, 34, 8, 1, 6, 26, 75, 125, 89, 13, 1, 7, 37, 139, 338, 450, 233, 21, 1, 8, 50, 233, 757, 1541, 1625, 610, 34, 1, 9, 65, 363, 1490, 4172, 7069, 5875, 1597, 55, 1, 10, 82, 535, 2669, 9633, 23165, 32532, 21250, 4181, 89

Offset: 0

Paul Barry, Mar 22 2003

Array rows are solutions of the recurrence a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2), where a(0) = 1 and a(1) = k+1.

			The array rows begins as:
  1, 1,  2,   3,    5,     8,     13, ... A000045;
  1, 2,  5,  13,   34,    89,    233, ... A001519;
  1, 3, 10,  35,  125,   450,   1625, ... A081567;
  1, 4, 17,  75,  338,  1541,   7069, ... A081568;
  1, 5, 26, 139,  757,  4172,  23165, ... A081569;
  1, 6, 37, 233, 1490,  9633,  62753, ... A081570;
  1, 7, 50, 363, 2669, 19814, 148153, ... A081571;
Antidiagonal triangle begins as:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,   3;
  1, 4, 10,  13,   5;
  1, 5, 17,  35,  34,    8;
  1, 6, 26,  75, 125,   89,   13;
  1, 7, 37, 139, 338,  450,  233,  21;
  1, 8, 50, 233, 757, 1541, 1625, 610, 34;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Array row n: A000045 (n=0), A001519 (n=1), A081567 (n=2), A081568 (n=3), A081569 (n=4), A081570 (n=5), A081571 (n=6).
Array column k: A000027 (k=1), A002522 (k=2).
Different from A073133.
Cf. A028387.

A081572:= func< n,k | (&+[Binomial(k,j)*Fibonacci(j+1)*(n-k)^(k-j): j in [0..k]]) >;
[A081572(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

T[n_, k_]:= If[n==0, Fibonacci[k+1], Sum[Binomial[k, j]*Fibonacci[j+1]*n^(k-j), {j, 0, k}]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2021 *)

def A081572(n,k): return sum( binomial(k,j)*fibonacci(j+1)*(n-k)^(k-j) for j in (0..k) )
flatten([[A081572(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Rows are successive binomial transforms of F(n+1).
T(n, k) = ((5+sqrt(5))/10)*( (2*n + 1 + sqrt(5))/2)^k + ((5-sqrt(5)/10)*( 2*n + 1 - sqrt(5))/2 )^k.
From G. C. Greubel, May 27 2021: (Start)
T(n, k) = Sum_{j=0..k} binomial(k,j)*n^(k-j)*Fibonacci(j+1) (square array).
T(n, k) = Sum_{j=0..k} binomial(k,j)*(n-k)^(k-j)*Fibonacci(j+1) (antidiagonal triangle). (End)

A111776 Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element chain) of waist k (waist(alpha) = max(Im(alpha))).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 6, 10, 1, 8, 12, 20, 35, 1, 16, 24, 40, 70, 125, 1, 32, 48, 80, 140, 250, 450, 1, 64, 96, 160, 280, 500, 900, 1625

Offset: 0

Abdullahi Umar, Aug 25 2008

G(n, n) is A081567(n - 1)

			G(3,2) = 6 because there are exactly 6 idempotent order-preserving partial transformations (on a 3-element chain) of waist 2, namely: (2)->(2), (1,2)->(1,2), (1,2)->(2,2),(1,3)->(3,3), (2,3)->(2,2), (2,3)->(3,3) - the mappings are coordinate-wise

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359.

Cf. A081567.

G(n,k)= (2^(n-k))*G(n,n)=(2^(n-k))*A081567(n-1), G(0,0) = 1

A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

Original entry on oeis.org

0, 0, 0, 1, 5, 22, 91, 361, 1392, 5265, 19653, 72694, 267179, 977593, 3565600, 12975457, 47142021, 171075606, 620303547, 2247803785, 8141857808, 29481675889, 106728951109, 386314552438, 1398132674955, 5059626441177, 18308871648576, 66249898660801

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A208737, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0, 0}, LinearRecurrence[{8, -21, 20, -5}, {0, 1, 5, 22}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:5,5:22}):
    if n in d:
        return d[n]
    d[n]=8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4)
    return d[n]

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), a(2) = 0, a(3) = 1, a(4) = 5, a(5) = 22.
G.f.: (x^3 - 3*x^4 + 3*x^5)/(1 - 8*x + 21*x^2 - 20*x^3 + 5*x^4); (x^3 * (1 - 3*x + 3*x^2))/((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)) .
a(n) = A081567(n-2) - A001519(n-1).

A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

Original entry on oeis.org

0, 0, 0, 1, 7, 37, 175, 778, 3325, 13837, 56524, 227866, 909832, 3607294, 14227447, 55894252, 218937532, 855650749, 3338323915, 13007422705, 50631143323, 196928737582, 765495534433, 2974251390529, 11552064922624, 44856304154086

Offset: 0

David Nacin, Mar 01 2012

Uniform used in the sense of Retakh, Serconek and Wilson. We use Stanley's definition of graded poset: all maximal chains have the same length n (which also implies all maximal elements have maximal rank.)

R. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (10,-36,57,-39,9).

Cf. A208736, A206901, A206902, A206947-A206950, A001906, A025192, A081567, A124302, A124292, A088305, A086405, A012781.

Join[{0}, LinearRecurrence[{10, -36, 57, -39, 9}, {0, 0, 1, 7, 37}, 40]]

def a(n, d={0:0,1:0,2:0,3:1,4:7,5:37}):
    if n in d:
        return d[n]
    d[n]=10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5)
    return d[n]

a(n) = 10*a(n-1) - 36*a(n-2) + 57*a(n-3) - 39*a(n-4) + 9*a(n-5), a(1) = 0, a(2) = 0, a(3) = 1, a(4) = 7, a(5) = 37.
G.f: (x^3 - 3*x^4 + 3*x^5)/(1 - 10*x + 36*x^2 - 57*x^3 + 39*x^4 - 9*x^5); (x^3*(1 - 3*x + 3*x^2)) / ((1 - x) (1 - 3*x) (1 - 6*x + 9*x^2 - 3*x^3)).
a(n) = A124292(n) - A124302(n).

A106198 Triangle, columns = successive binomial transforms of Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 13, 10, 4, 1, 8, 34, 35, 17, 5, 1, 13, 89, 125, 75, 26, 6, 1, 21, 233, 450, 338, 139, 37, 7, 1, 34, 610, 1625, 1541, 757, 233, 50, 8, 1

Offset: 0

Gary W. Adamson, Apr 24 2005

Column 0 = Fibonacci numbers, column 1 = odd-indexed Fibonacci numbers (first binomial transform of 1, 1, 2, 3, 5, ...); column 2 = second binomial transform of Fibonacci numbers, etc.

			First few rows of the triangle are:
   1;
   1,   1;
   2,   2,   1;
   3,   5,   3,   1;
   5,  13,  10,   4,   1;
   8,  34,  35,  17,   5,   1;
  13,  89, 125,  75,  26,   6,   1;
  21, 233, 450, 338, 139,  37,   7,   1;
  ...
Column 2 = A081567, second binomial transform of Fibonacci numbers: 1, 3, 10, 35, 125, ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000045, A081567, A081568, A081569, A081570.

T:= function(n,k)
    if k=0 then return Fibonacci(n+1);
    else return Sum([0..n-k], j-> Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j));
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Dec 11 2019

function T(n,k)
  if k eq 0 then return Fibonacci(n+1);
  else return (&+[Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j): j in [0..n-k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 11 2019

with(combinat);
T:= proc(n, k) option remember;
      if k=0 then fibonacci(n+1)
    else add( binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j), j=0..n-k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Dec 11 2019

Table[If[k==0, Fibonacci[n+1], Sum[Binomial[n-k, j]*Fibonacci[j+1]*k^(n-k-j), {j,0,n-k}]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 11 2019 *)

T(n,k) = if(k==0, fibonacci(n+1), sum(j=0,n-k, binomial(n-k,j)*fibonacci( j+1)*k^(n-k-j)) ); \\ G. C. Greubel, Dec 11 2019

@CachedFunction
def T(n, k):
    if (k==0): return fibonacci(n+1)
    else: return sum(binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 11 2019

Offset column k = k-th binomial transform of the Fibonacci numbers, given leftmost column = Fibonacci numbers.

A214582 Riordan array (1/(1-x-x^2), x(1+2x)).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 4, 5, 1, 5, 7, 10, 7, 1, 8, 11, 15, 20, 9, 1, 13, 18, 25, 35, 34, 11, 1, 21, 29, 40, 55, 75, 52, 13, 1, 34, 47, 65, 90, 125, 143, 74, 15, 1, 55, 76, 105, 145, 200, 275, 247, 100, 17, 1

Offset: 0

Philippe Deléham, Mar 06 2013

First column is A000045 (Fibonacci numbers) starting with 1.

Second column is A000032 (Lucas numbers) starting with 1.

			Triangle begins
1
1, 1
2, 3, 1
3, 4, 5, 1
5, 7, 10, 7, 1
8, 11, 15, 20, 9, 1
13, 18, 25, 35, 34, 11, 1
21, 29, 40, 55, 75, 52, 13, 1
34, 47, 65, 90, 125, 143, 74, 15, 1
55, 76, 105, 145, 200, 275, 247, 100, 17, 1
...
Production array begins
1, 1
1, 2, 1
-2, -4, 2, 1
8, 16, -4, 2, 1
-40, -80, 16, -4, 2, 1
224, 448, -80, 16, -4, 2, 1
-1344, -2688, 448, -80, 16, -4, 2, 1
8448, 16896, -2688, 448, -80, 16, -4, 2, 1
... which is based on A052701.

Cf. A000032, A000045, A002522, A005408, A104726, A123265

T(n,0) = T(n-1,0) + T(n-2,0), T(n,k) = T(n-1,k-1) + 2*T(n-2,k-1) for k>0.
Sum_{k, 0<=k<=n} T(n,k) = A094687(n+2).
T(2n,n) = A081567(n).

A260304 a(n) = 5a(n-1) - 5a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 5, 10, 25, 75, 250, 875, 3125, 11250, 40625, 146875, 531250, 1921875, 6953125, 25156250, 91015625, 329296875, 1191406250, 4310546875, 15595703125, 56425781250, 204150390625, 738623046875, 2672363281250, 9668701171875, 34981689453125, 126564941406250

Offset: 0

Ilya Gutkovskiy, Nov 21 2015

Lim_{n -> infinity} a(n + 1)/a(n) = 2 + phi = 3.6180339887..., where phi is the golden ratio (A001622).

Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A020876, A030191, A098111.

Cf. A093129: initial values 1,2; A081567: initial values 1,3.

[n le 2 select n+1 else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 23 2015

Table[((5 + 2 Sqrt[5]) ((5 - Sqrt[5])/2)^n + (5 - 2 Sqrt[5]) ((5 + Sqrt[5])/2)^n)/5, {n, 0, 30}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 5 a[n - 1] - 5 a[n - 2]}, a, {n, 0, 30}] (* Bruno Berselli, Nov 23 2015 *)

a(n)=([0,1; -5,5]^n*[2;3])[1,1] \\ Charles R Greathouse IV, Jul 26 2016

G.f.: (2 - 7*x)/(1 - 5*x + 5*x^2).
a(n) = ((5 + 2*sqrt(5))*((5 - sqrt(5))/2)^n + (5 - 2*sqrt(5))*((5 + sqrt(5))/2)^n)/5.
a(n) = 2*A030191(n) - 7*A030191(n-1). - Bruno Berselli, Nov 23 2015

Edited by Bruno Berselli, Nov 23 2015

A376499 Array read by ascending antidiagonals: A(n,k) = A376484/(2*n+1).

Original entry on oeis.org

1, 1, 3, 1, 3, 9, 1, 3, 10, 27, 1, 3, 10, 35, 81, 1, 3, 10, 35, 125, 243, 1, 3, 10, 35, 126, 450, 729, 1, 3, 10, 35, 126, 462, 1625, 2187, 1, 3, 10, 35, 126, 462, 1715, 5875, 6561, 1, 3, 10, 35, 126, 462, 1716, 6419, 21250, 19683, 1, 3, 10, 35, 126, 462, 1716, 6435, 24157, 76875, 59049

Offset: 1

Cheng-Jun Li, Sep 25 2024

It is only a conjecture that the A(n,k) are always integers.

Values repeated as a staircase for all A(n+x,2*n) (x > 0 and are equal to A(n,2*n)).

			First ten rows start as follows:
  1 3  9 27  81 243  729 2187  6561 19683  59049  177147  531441  1594323  4782969
  1 3 10 35 125 450 1625 5875 21250 76875 278125 1006250 3640625 13171875 47656250
  1 3 10 35 126 462 1715 6419 24157 91238 345401 1309574 4970070 18874261 71705865
  1 3 10 35 126 462 1716 6435 24309 92358 352485 1350054 5185350 19960020 76964985
  1 3 10 35 126 462 1716 6435 24310 92378 352715 1352054 5199975 20055024 77531355
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200299 20058272 77558325
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558759
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760

All of these are conjectures. Rows: A000244, A081567, A122068. Columns: A000012, A000012 * 3, A095049 for n >= 20. A(1,k) = A000244, A(2,k) = A081567, A(3,k) = A122068 (First 3 rows of the array).A(n,1) = A(n,2) / 3 = A000012, A(n,3) = A095049 for n >= 20 (First 3 columns of the array). When k increases, the row of A(n,k) gets closer to A001700.

A(n,k) = A376484(n,k)/(2*n+1)

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A081572 Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.

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A111776 Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element chain) of waist k (waist(alpha) = max(Im(alpha))).

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A208736 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with exactly 2 elements of each rank level between 0 and 1.

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A208737 Number of nonisomorphic graded posets with 0 and 1 and non-uniform Hasse graph of rank n, with no 3-element antichain.

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A106198 Triangle, columns = successive binomial transforms of Fibonacci numbers.

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A214582 Riordan array (1/(1-x-x^2), x*(1+2*x)).

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A260304 a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.

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A214582 Riordan array (1/(1-x-x^2), x(1+2x)).

A260304 a(n) = 5a(n-1) - 5a(n-2) for n>1, a(0)=2, a(1)=3.