A067411 -id:A067411 - cp's OEIS frontend

A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

1, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -2, 0, 2, 1, -1, -3, -2, 2, 3, 1, -1, -4, -5, 0, 5, 4, 1, -1, -5, -9, -5, 5, 9, 5, 1, -1, -6, -14, -14, 0, 14, 14, 6, 1, -1, -7, -20, -28, -14, 14, 28, 20, 7, 1, -1, -8, -27, -48, -42, 0, 42, 48, 27, 8, 1, -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1, -1, -10, -44, -110, -165, -132, 0, 132, 165, 110

Offset: 0

Paul Barry, Sep 06 2005

Row sums are A000007. Diagonal sums are -F(n-2). Inverse is A112468. T(2n,n)=0.
(-1,1)-Pascal triangle. - Philippe Deléham, Aug 07 2006
Apart from initial term, same as A008482. - Philippe Deléham, Nov 07 2006
Each column equals the cumulative sum of the previous column. - Mats Granvik, Mar 15 2010
Reading along antidiagonals generates in essence rows of A192174. - Paul Curtz, Oct 02 2011
Triangle T(n,k), read by rows, given by (-1,2,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011

			Triangle starts:
    1;
   -1,  1;
   -1,  0,   1;
   -1, -1,   1,   1;
   -1, -2,   0,   2,   1;
   -1, -3,  -2,   2,   3,   1;
   -1, -4,  -5,   0,   5,   4,  1;
   -1, -5,  -9,  -5,   5,   9,  5,  1;
   -1, -6, -14, -14,   0,  14, 14,  6,  1;
   -1, -7, -20, -28, -14,  14, 28, 20,  7,  1;
   -1, -8, -27, -48, -42,   0, 42, 48, 27,  8, 1;
   -1, -9, -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
  ...
From _Paul Barry_, Apr 08 2011: (Start)
Production matrix begins:
   1,  1,
  -2, -1,  1,
   2,  0, -1,  1,
  -2,  0,  0, -1,  1,
   2,  0,  0,  0, -1,  1,
  -2,  0,  0,  0,  0, -1,  1,
   2,  0,  0,  0,  0,  0, -1,  1
  ... (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Restricting Dyck Paths and 312-avoiding Permutations, arXiv:2307.02837 [math.CO], 2023. Mentions this sequence.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
Emeric Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
D. Foata and G.-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.
Jack Ramsay, On Arithmetical Triangles, The Pulse of Long Island, June 1965. [Mentions application to design of antenna arrays. Annotated scan.]

Same first 3 rows as in A054525.

Cf. A008482, A037012, A080232, A112466, A112467, A174293, A174294, A174295, A174296, A174297.

[n eq 0 select 1 else (2*k-n)*Binomial(n,k)/n: k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 04 2019

seq(seq( `if`(n=0, 1, (2*k-n)*binomial(n,k)/n), k=0..n), n=0..10); # G. C. Greubel, Dec 04 2019

T[n_, k_]= If[n==0, 1, ((2*k-n)/n)*Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019 *)

T(n, k) = if(n==0, 1, (2*k-n)*binomial(n,k)/n ); \\ G. C. Greubel, Dec 04 2019

def T(n, k):
    if (n==0): return 1
    else: return (2*k-n)*binomial(n,k)/n
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 04 2019

Number triangle T(n, k) = binomial(n, n-k) - 2*binomial(n-1, n-k-1).
Sum_{k=0..n} T(n, k)*x^k = (x-1)*(x+1)^(n-1). - Philippe Deléham, Oct 03 2005
T(n,k) = ((2*k-n)/n)*binomial(n, k), with T(0,0)=1. - Roger L. Bagula, Feb 16 2009; modified by G. C. Greubel, Dec 04 2019
T(n,k) = T(n-1,k-1) + T(n-1,k) with T(0,0)=1, T(1,0)=-1, T(n,k)=0 for k>n or for n<0. - Philippe Deléham, Nov 01 2011
G.f.: (1-2x)/(1-(1+y)*x). - Philippe Deléham, Dec 15 2011
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A133494(n), A081294(n), A005053(n), A067411(n), A199661(n), A083233(n) for x = 1, 2, 3, 4, 5, 6, 7, respectively. - Philippe Deléham, Dec 15 2011
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(-1 - x + x^2/2! + x^3/3!) = -1 - 2*x - 2*x^2/2! + 5*x^4/4! + 14*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014
Sum_{k=0..n} T(n,k) = 0^n = A000007(n). - G. C. Greubel, Dec 04 2019

A067410 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 12, 4, 1, 16, 48, 24, 5, 1, 32, 192, 144, 40, 6, 1, 64, 768, 864, 320, 60, 7, 1, 128, 3072, 5184, 2560, 600, 84, 8, 1, 256, 12288, 31104, 20480, 6000, 1008, 112, 9, 1, 512, 49152, 186624, 163840, 60000, 12096, 1568, 144, 10, 1

Offset: 0

Wolfdieter Lang, Jan 25 2002

			Triangle starts:
  1;
  2,  1;
  4,  3, 1;
  8, 12, 4, 1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998 (triangle built from powers of (m+1)), A067402.

Columns m=0..8 are A000079, A002001, A067411, A067412, A090019, A067413, A067414, A067415, A067416.

A[n_,m_]:=If[n==m,1,(m+2)(2(m+1))^(n-m-1)]; Flatten[Table[A[n,m],{n,0,9},{m,0,n}]] (* Stefano Spezia, Sep 30 2022 *)

a(n, m) = 1 if n = m; a(n, m) = (m+2)*(2*(m+1))^(n-m-1) if n > m >= 0.

G.f. for column m: (x^m)*(1-m*x)/(1-2*(m+1)*x).

A067417 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 24, 5, 1, 81, 144, 45, 6, 1, 243, 864, 405, 72, 7, 1, 729, 5184, 3645, 864, 105, 8, 1, 2187, 31104, 32805, 10368, 1575, 144, 9, 1, 6561, 186624, 295245, 124416, 23625, 2592, 189, 10, 1, 19683, 1119744, 2657205, 1492992, 354375, 46656, 3969, 240, 11, 1

Offset: 0

Wolfdieter Lang, Jan 25 2002

			Triangle starts:
   1;
   3,  1;
   9,  4, 1;
  27, 24, 5, 1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998 (triangle built from powers of (m+1)), A067402, A067410.

Columns m=0..8 are A000244, A067411, A067403, A067419, A067420, A067421, A067422, A067423, A067424.

A[n_,m_]:=If[n==m,1,(m+3)(3(m+1))^(n-m-1)]; Flatten[Table[A[n,m],{n,0,9},{m,0,n}]] (* Stefano Spezia, Sep 30 2022 *)

a(n, m) = 1 if n = m; a(n, m) = (m+3)*(3*(m+1))^(n-m-1) if n > m >= 0.

G.f. for column m: (x^m)*(1-2*m*x)/(1-3*(m+1)*x).

A084477 Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.

Original entry on oeis.org

4, 2, 8, 48, 288, 1728, 10368, 62208, 373248, 2239488, 13436928, 80621568, 483729408, 2902376448, 17414258688, 104485552128, 626913312768, 3761479876608, 22568879259648, 135413275557888, 812479653347328, 4874877920083968, 29249267520503808

Offset: 1

Ralf Stephan, May 27 2003

A tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017

Fault-free tilings are those where the only straight interface is at the left and right end. Thus a(n) <= A046984(n).

Colin Barker, Table of n, a(n) for n = 1..1000
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles
C. Moore, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A084478, A084479, A084480, A084481.

Vec(2*x*(2 - 11*x - 2*x^2) / (1 - 6*x) + O(x^30)) \\ Colin Barker, Mar 28 2017

a(n) = 2*A067411(n-2) for n>1.
G.f.: 2*z(2-11*z-2*z^2) / (1-6*z).
a(n) = 8 * 6^(n-3) for n>2.
G.f.: 9/2 - x - 1/Q(0) where Q(k)= 1 + 5^k/(1 - 2*x/(2*x + 5^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013
a(n) = 6*a(n-1) for n>2. - Colin Barker, Mar 28 2017

A067412 Fourth column of triangle A067410.

Original entry on oeis.org

1, 5, 40, 320, 2560, 20480, 163840, 1310720, 10485760, 83886080, 671088640, 5368709120, 42949672960, 343597383680, 2748779069440, 21990232555520, 175921860444160, 1407374883553280, 11258999068426240

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column gives [1,6,60,600,6000,60000,...].

a(n+1) = A157176(A016957(n)). [From Reinhard Zumkeller, Feb 24 2009]

Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A067411 (third column), A067413 (sixth column), A001018 (powers of 8).

Join[{1},NestList[8#&,5,20]] (* or *) CoefficientList[Series[ (1-3x)/ (1-8x),{x,0,20}],x] (* Harvey P. Dale, May 14 2011 *)

a(n)= A067410(n+3, 3). a(n)= 5*8^(n-1), n>=1, a(0)=1.
G.f.: (1-3*x)/(1-8*x).
E.g.f.: (5*exp(8*x)+3)/8 = exp(4*x)*(cosh(4*x)+sinh(4*x)/4) - Paul Barry, Nov 20 2003

A083076 Third row of number array A083075.

Original entry on oeis.org

1, 5, 33, 229, 1601, 11205, 78433, 549029, 3843201, 26902405, 188316833, 1318217829, 9227524801, 64592673605, 452148715233, 3165041006629, 22155287046401, 155087009324805, 1085609065273633, 7599263456915429, 53194844198408001

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A067411. Inverse binomial transform of A082412.

Trinomial transform of Jacobsthal numbers A001045. - Paul Barry, Sep 10 2007

Nathaniel Johnston, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A034478, A066443, A082761, A083077.

[(2*7^n+1)/3: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

seq((2*7^n+1)/3,n=0..20); # Nathaniel Johnston, Jun 26 2011

a(n) = (2*7^n + 1)/3.
G.f.: (1-3*x)/((1-x)*(1-7*x)).
E.g.f.: (2*exp(7*x) + exp(x))/3.
a(n) = Sum_{k=0..2*n} trinomial(n,k)*Fibonacci(k+1), where trinomial(n,k) are the trinomial coefficients (A027907). - Paul Barry, Sep 10 2007
a(n) = 7*a(n-1) - 2, a(n) = 8*a(n-1) - 7*a(n-2). - Vincenzo Librandi, Nov 06 2011

A203984 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having equal diagonal elements or equal antidiagonal elements, and new values 0..2 introduced in row major order.

Original entry on oeis.org

6, 24, 24, 96, 144, 96, 384, 864, 864, 384, 1536, 5184, 7776, 5184, 1536, 6144, 31104, 69984, 69984, 31104, 6144, 24576, 186624, 629856, 956448, 629856, 186624, 24576, 98304, 1119744, 5668704, 13071456, 13071456, 5668704, 1119744, 98304, 393216

Offset: 1

R. H. Hardin Jan 09 2012

Table starts
.....6......24........96.........384..........1536............6144
....24.....144.......864........5184.........31104..........186624
....96.....864......7776.......69984........629856.........5668704
...384....5184.....69984......956448......13071456.......178855776
..1536...31104....629856....13071456.....271918944......5671161216
..6144..186624...5668704...178855776....5671161216....180709558848
.24576.1119744..51018336..2447270496..118333620576...5764846339584
.98304.6718464.459165024.33489653472.2469841766784.184042295652096

			Some solutions for n=4 k=3
..0..0..0..0....0..0..0..0....0..0..1..1....0..1..0..0....0..1..2..1
..1..1..1..2....1..2..1..2....1..2..2..0....2..2..2..1....2..1..2..0
..0..0..0..2....1..2..1..0....0..0..1..1....0..0..0..0....2..0..2..1
..1..1..1..1....0..2..1..2....1..2..2..0....1..1..1..1....1..1..2..1
..0..2..2..2....1..2..1..2....1..0..1..1....2..2..2..2....2..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A002023(n-1)

Column 2 is A067411(n+1)

Empirical for column k:
k=1: a(n) = 6*4^(n-1)
k=2: a(n) = 4*6^n
k=3: a(n) = 96*9^(n-1)
k=4: a(n) = 15*a(n-1) -270*a(n-3) +324*a(n-4)
k=5: a(n) = 25*a(n-1) -45*a(n-2) -963*a(n-3) +2025*a(n-4) +3645*a(n-5) -6561*a(n-6)
k=6: (order 15 recurrence)
k=7: (order 45 recurrence)

A164532 a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 6, 24, 36, 144, 216, 864, 1296, 5184, 7776, 31104, 46656, 186624, 279936, 1119744, 1679616, 6718464, 10077696, 40310784, 60466176, 241864704, 362797056, 1451188224, 2176782336, 8707129344, 13060694016, 52242776064, 78364164096

Offset: 1

Klaus Brockhaus, Aug 15 2009

nonn

Interleaving of A000400 and A067411 without initial term 1.

Binomial transform is apparently A123011. Fourth binomial transform is A154235.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A000400 (powers of 6), A067411, A123011, A154235.

[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..29] ];

LinearRecurrence[{0,6}, {1,4}, 40] (* G. C. Greubel, Jul 16 2021 *)

[((1 - (-1)^n)*sqrt(6)/2 + 2*(1 + (-1)^n))*6^(n/2 -1) for n in (1..40)] # G. C. Greubel, Jul 16 2021

a(n) = (5 - (-1)^n)*6^(1/4*(2*n - 5 + (-1)^n)).
G.f.: x*(1+4*x)/(1-6*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = ((1-(-1)^n)*sqrt(6)/2 + 2*(1+(-1)^n))*6^(n/2 -1). - G. C. Greubel, Jul 16 2021

A083217 a(n) = (2*5^n + (-1)^n)/3.

Original entry on oeis.org

1, 3, 17, 83, 417, 2083, 10417, 52083, 260417, 1302083, 6510417, 32552083, 162760417, 813802083, 4069010417, 20345052083, 101725260417, 508626302083, 2543131510417, 12715657552083, 63578287760417, 317891438802083

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A003683 (without leading zero). Inverse binomial transform of A067411.

a(n) is the number of compositions of n when there are 3 types of 1 and 8 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A003683, A067411, A082412.

[(2*5^n +(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Feb 17 2023

LinearRecurrence[{4,5},{1,3},30] (* Harvey P. Dale, Sep 18 2018 *)

from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1,3,4,5, lambda n: 0)
[next(it) for i in range(1,24)] # Zerinvary Lajos, Jul 03 2008

a(n) = (2*5^n + (-1)^n)/3.
G.f.: (1-x)/((1-5*x)*(1+x)).
E.g.f.: (2*exp(5*x) + exp(-x))/3
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*J(n-j+1) where J(n) = A001045(n). - Paul Barry, May 19 2006
a(0)=1, a(n) = 5*a(n-1) - 2 if n is odd, and a(n) = 5*a(n) + 2 if n is even. - Vincenzo Librandi, Nov 18 2010

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)b(i-1,j)-c(i,j)c(i,j-1) is nonzero.

Original entry on oeis.org

36, 144, 144, 576, 864, 576, 2304, 5184, 5184, 2304, 9216, 31104, 46656, 31104, 9216, 36864, 186624, 419904, 419904, 186624, 36864, 147456, 1119744, 3779136, 5738688, 3779136, 1119744, 147456, 589824, 6718464, 34012224, 78428736, 78428736

Offset: 1

R. H. Hardin Jan 10 2012

Also 0..2 arrays with no 2X2 subblock having equal diagonal elements or equal antidiagonal elements
Table starts
.....36......144........576.........2304...........9216............36864
....144......864.......5184........31104.........186624..........1119744
....576.....5184......46656.......419904........3779136.........34012224
...2304....31104.....419904......5738688.......78428736.......1073134656
...9216...186624....3779136.....78428736.....1631513664......34026967296
..36864..1119744...34012224...1073134656....34026967296....1084257353088
.147456..6718464..306110016..14683622976...710001723456...34589078037504
.589824.40310784.2754990144.200937920832.14819050600704.1104253773912576

			Some solutions for n=5 k=3
..0..1..0..1....1..2..0..1....0..1..2..1....2..2..0..1....2..2..2..1
..2..1..2..1....1..2..0..1....2..1..0..0....0..1..0..2....0..0..0..1
..0..0..0..1....1..2..0..2....2..1..2..1....2..1..0..1....2..2..2..2
..1..1..2..1....1..2..0..1....2..0..2..0....2..1..2..2....0..0..0..1
..0..0..2..0....1..2..0..1....1..0..2..0....0..0..0..0....2..2..2..2
..1..1..1..1....0..2..0..1....1..0..1..1....1..2..1..2....1..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..285

Column 1 is A002063
Column 2 is A067411(n+2)
Column 3 is A055995(n+2)

Empirical for column k:
k=1: T(n,k)=4*T(n-1,k)
k=2: T(n,k)=6*T(n-1,k)
k=3: T(n,k)=9*T(n-1,k)
k=4: T(n,k)=15*T(n-1,k)-270*T(n-3,k)+324*T(n-4,k)
k=5: T(n,k)=25*T(n-1,k)-45*T(n-2,k)-963*T(n-3,k)+2025*T(n-4,k)+3645*T(n-5,k)-6561*T(n-6,k)
k=6: (order 15)
k=7: (order 45)

cp's OEIS Frontend

A112467 Riordan array ((1-2x)/(1-x), x/(1-x)).

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A067410 Triangle with columns built from certain power sequences.

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A067417 Triangle with columns built from certain power sequences.

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A084477 Number of fault-free tilings of a 4 X 3n rectangle with right trominoes.

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A067412 Fourth column of triangle A067410.

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A083076 Third row of number array A083075.

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A203984 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having equal diagonal elements or equal antidiagonal elements, and new values 0..2 introduced in row major order.

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A164532 a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4.

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A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)b(i-1,j)-c(i,j)c(i,j-1) is nonzero.