A085478 -id:A085478 - cp's OEIS frontend

A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

1, 2, 1, 4, 5, 1, 8, 18, 8, 1, 16, 56, 41, 11, 1, 32, 160, 170, 73, 14, 1, 64, 432, 620, 377, 114, 17, 1, 128, 1120, 2072, 1666, 704, 164, 20, 1, 256, 2816, 6496, 6608, 3649, 1178, 223, 23, 1, 512, 6912, 19392, 24192, 16722, 7001, 1826, 291, 26, 1, 1024, 16640, 55680, 83232, 69876, 36365, 12235, 2675, 368, 29, 1

Offset: 0

Paul Barry, Nov 15 2005

Row sums are A081567. Diagonal sums are A085810. Product of Pascal triangle A007318 and Morgan-Voyce triangle A085478.

Unsigned version of A123876. - Philippe Deléham, Oct 25 2007

			Triangle begins:
   1;
   2,   1;
   4,   5,   1;
   8,  18,   8,  1;
  16,  56,  41, 11,  1;
  32, 160, 170, 73, 14, 1;
  ...

Cf. A081567, A085810, A007318, A085478, A123876.

T(2n,n) gives A026000.

Number triangle T(n,k) = Sum_{j=0..n} C(n, j)*C(j+k, 2k);
T(n,k) = Sum_{j=0..n} C(n, k+j)*C(k, k-j)*2^(n-k-j);
T(n,k) = Sum_{j=0..n-k} C(n+k-j, n-k-j)*C(k, j)*(-1)^j*2^(n-k-j).
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 17 2014

More terms from Michel Marcus, Sep 09 2024

A123878 Product of signed and unsigned Morgan-Voyce triangles.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -1, -3, 0, 1, 0, -3, -5, 0, 1, 1, 3, -5, -7, 0, 1, 1, 9, 10, -7, -9, 0, 1, 0, 5, 25, 21, -9, -11, 0, 1, -1, -9, 5, 49, 36, -11, -13, 0, 1, -1, -18, -50, -7, 81, 55, -13, -15, 0, 1, 0, -7, -70, -147, -39, 121, 78, -15, -17, 0, 1

Offset: 0

Paul Barry, Oct 16 2006

Inverse is A123880.

Row sums are A123879.

			Number triangle begins:
   1;
   0,  1;
  -1,  0,  1;
  -1, -3,  0,  1;
   0, -3, -5,  0,  1;
   1,  3, -5, -7,  0, 1;
   1,  9, 10, -7, -9, 0, 1;

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

Cf. A085478, A123879, A123880.

B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(j-k)*B(n+j,2*j)*B(n+j,2*k) )))); # G. C. Greubel, Aug 08 2019

B:= Binomial; [(&+[(-1)^(j-k)*B(n+j,2*j)*B(n+j,2*k):j in [0..n]]) : k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 08 2019

Table[Sum[(-1)^(j-k)*Binomial[n+j,2*j]*Binomial[j+k,2*k], {j,0,n}], {n, 0, 12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 08 2019 *)

T(n,k) = sum(j=0,n, (-1)^(j-k)*binomial(n+j,2*j)*binomial(n+j,2*k) );
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 08 2019

b=binomial; [[sum((-1)^(j-k)*b(n+j,2*j)*b(n+j,2*k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 08 2019

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

Number triangle: T(n,k) = Sum_{j=0..n} C(n+j,2*j)*C(j+k,2*k)*(-1)^(j-k).

A123970 Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the n X n matrix (min(i,j)) (i,j=1,2,...,n) (0 <= k <= n, n >= 1).

Original entry on oeis.org

1, 1, -1, 1, -3, 1, 1, -6, 5, -1, 1, -10, 15, -7, 1, 1, -15, 35, -28, 9, -1, 1, -21, 70, -84, 45, -11, 1, 1, -28, 126, -210, 165, -66, 13, -1, 1, -36, 210, -462, 495, -286, 91, -15, 1, 1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1, 1, -55, 495, -1716, 3003, -3003, 1820, -680, 153, -19, 1, 1, -66, 715, -3003, 6435, -8008

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 29 2006

This sequence is the same as A129818 up to sign. - T. D. Noe, Sep 30 2011

Riordan array (1/(1-x), -x/(1-x)^2). - Philippe Deléham, Feb 18 2012

			Triangular sequence (gives the odd Tutte-Beraha constants as roots!) begins:
  1;
  1,  -1;
  1,  -3,   1;
  1,  -6,   5,   -1;
  1, -10,  15,   -7,    1;
  1, -15,  35,  -28,    9,    -1;
  1, -21,  70,  -84,   45,   -11,   1;
  1, -28, 126, -210,  165,   -66,  13,   -1;
  1, -36, 210, -462,  495,  -286,  91,  -15,  1;
  1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1;
  ...

S. Beraha, Infinite non-trivial families of maps and chromials, Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1975.
Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), chapter 5.25.
W. T. Tutte, "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.

Paul Fendley and Vyacheslav Krushkal, Tutte chromatic identities and the Temperley-Lieb algebra, arXiv:0711.0016 [math.CO], 2007-2008.
Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. See page 161.
Eric Weisstein's World of Mathematics, Beraha Constants

Cf. A085478, A076756.
Cf. A109954, A129818, A143858, A165253. - R. J. Mathar, Jan 10 2011
Modulo signs, inverse matrix to A039599.

/* As triangle */ [[(-1)^k*Binomial(n + k, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jan 04 2019

with(linalg): m:=(i,j)->min(i,j): M:=n->matrix(n,n,m): T:=(n,k)->coeff(charpoly(M(n),x),x,n-k): 1; for n from 1 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

An[d_] := MatrixPower[Table[Min[n, m], {n, 1, d}, {m, 1, d}], -1]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]

f(n,x) = (2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the characteristic polynomial of the n X n matrix from the definition and f(0,x)=1. See formula in Fendley and Krushkal. - Jonathan Vos Post, Nov 04 2007
T(n,k) = (-1)^k * A085478(n,k) = (-1)^n * A129818(n,k). - Philippe Deléham, Feb 06 2012
T(n,k) = 2*T(n-1,k) - T(n-1,k-1) - T(n-2,k), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 29 2013

Edited by N. J. A. Sloane, Nov 29 2006

A190909 Triangle read by rows: T(n,k) = binomial(n+k,n-k) * k! / floor(k/2)!^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 6, 1, 15, 70, 168, 54, 30, 1, 21, 140, 504, 270, 330, 20, 1, 28, 252, 1260, 990, 1980, 260, 140, 1, 36, 420, 2772, 2970, 8580, 1820, 2100, 70, 1, 45, 660, 5544, 7722, 30030, 9100, 16800, 1190, 630

Offset: 0

Peter Luschny, May 24 2011

The triangle may be regarded as a generalization of the triangle A063007.
A063007(n,k) = binomial(n+k, n-k)*(2*k)$;
T(n,k) = binomial(n+k, n-k)*(k)$.
Here n$ denotes the swinging factorial A056040(n). As A063007 is a decomposition of the central Delannoy numbers A001850, a combinatorial interpretation of T(n,k) in terms of lattice paths can be expected.
T(n,n) = A056040(n) which can be seen as extended central binomial numbers.

			[0]  1
[1]  1,  1
[2]  1,  3,   2
[3]  1,  6,  10,    6
[4]  1, 10,  30,   42,   6
[5]  1, 15,  70,  168,  54,   30
[6]  1, 21, 140,  504, 270,  330,  20
[7]  1, 28, 252, 1260, 990, 1980, 260, 140

Peter Luschny, The lost Catalan numbers
R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Cf. Row sums: A190910; A056040, A063007, A085478, A088617.

A190909 := (n,k) -> binomial(n+k,n-k)*k!/iquo(k,2)!^2:
seq(print(seq(A190909(n,k),k=0..n)),n=0..7);

Flatten[Table[Binomial[n+k,n-k] k!/(Floor[k/2]!)^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Mar 25 2012 *)

T(n,1) = A000217(n). T(n,2) = 2*binomial(n+2,4) (Cf. A034827).

A108367 L(n,-n), where L is defined as in A108299.

Original entry on oeis.org

1, -2, 5, -29, 265, -3191, 47321, -832040, 16908641, -389806471, 10049731549, -286482047279, 8946795882025, -303762892305614, 11140078609864049, -438857301101610929, 18482410314337295233, -828657053219851847135, 39406519321199703822581, -1981132660316876165976260

Offset: 0

Reinhard Zumkeller, Jun 01 2005

sign

A108366(n) = L(n,n).

Eric Weisstein's World of Mathematics, Morgan-Voyce polynomials

Cf. A000312, A085478, A108299, A108366.

a(n) = sum(k=0, n, (-1)^k*binomial(n+k,2*k)*(n+2)^k); \\ Jinyuan Wang, Feb 25 2020

a(n) = (-1)^n * Product_{k=1..n} (n + 2*cos((2*k-1)*Pi/(2*n+1))) with Pi = 3.14...

a(n) = Sum_{k=0..n} (-1)^k*binomial(n+k,2*k)*(n+2)^k = b(n,-n-2), where b(n,x) are the Morgan-Voyce polynomials of A085478. - Peter Bala, May 01 2012

More terms from Jinyuan Wang, Feb 25 2020

A114192 Riordan array (1/(1-2x),x/(1-2x)^2).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 24, 10, 1, 16, 80, 60, 14, 1, 32, 240, 280, 112, 18, 1, 64, 672, 1120, 672, 180, 22, 1, 128, 1792, 4032, 3360, 1320, 264, 26, 1, 256, 4608, 13440, 14784, 7920, 2288, 364, 30, 1, 512, 11520, 42240, 59136, 41184, 16016, 3640, 480, 34, 1

Offset: 0

Paul Barry, Nov 16 2005

Factors as (1/(1-x),x/(1-x))*(1/(1-x),x*(1+x)/(1-x)^2) or A007318 times A114188. Also (1/(1-2*x),x/(1-2*x))*(1,x*(1+2*x)). Inverse is A114193. Row sums are A007583. Diagonal sums are A007051.

			Triangle begins
1;
2, 1;
4, 6, 1;
8, 24, 10, 1;
16, 80, 60, 14, 1;
32, 240, 280, 112, 18, 1;

T(n,k) = sum{j=0..n, C(k, j)*C(n, k+j)}*2^(n-k).
T(n,k) = 2^(n-k)*binomial(n+k,2k) = 2^(n-k)*A085478(n,k). - Philippe Deléham, May 05 2006
T(n,k) = A013609(n+k, n-k). - Johannes W. Meijer, Sep 05 2013
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014

A144251 Eigensequence of triangle A054142.

Original entry on oeis.org

1, 1, 2, 6, 24, 122, 758, 5606, 48378, 479532, 5390940, 68022932, 954948752, 14804391270, 251815549396, 4673137101108, 94148342547146, 2050127343000170, 48061939075355080, 1208742383083994580, 32507565146820336836, 932149980847656487522, 28423646163259392354386, 919399182232129554488328

Offset: 0

Gary W. Adamson, Sep 16 2008

nonn

Eigensequence of the reversed triangle (A085478) = A125273.

Eigentriangle A144252 has row sums of A144251 shifted: (1, 2, 6, 24, 122,...) with right border = A144251.

			Triangle A054142 begins:
  1;
  1, 1;
  1, 3, 1;
  1, 5, 6, 1;
  1, 7, 15, 10, 1;
  1, 9, 28, 35, 15, 1;
  ...
a(3) = 6 = 1*1 + 3*1 + 1*2
a(4) = 24 = 1*1 + 5*1 + 6*2 + 1*6

Seiichi Manyama, Table of n, a(n) for n = 0..376

Cf. A054142, A085478, A144252,

A054142(n, k) = binomial(2*n-k, k);
a(n) = if (n==0, 1, sum(k=0, n-1, A054142(n-1,k)*a(k))); \\ too slow
lista(nn) = my(v=vector(nn)); v[1] = 1; for (n=2, nn, v[n] = sum(k=0, n-1, A054142(n-2,k)*v[k+1]);); v; \\ Michel Marcus, Jan 17 2025

a(n) = Sum_{k=0..n-1} A054142(n-1,k)*a(k) for n>0 with a(0)=1.

More terms from Seiichi Manyama, May 31 2022

A144252 Eigentriangle, row sums = A144251 shifted, right border = A144251.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 12, 6, 1, 7, 30, 60, 24, 1, 9, 56, 210, 360, 122, 1, 11, 90, 504, 1680, 2562, 758, 1, 13, 132, 990, 5040, 15372, 21224, 5606, 1, 15, 182, 1716, 11880, 36364, 159180, 201816, 47378, 1, 17, 240, 2730, 24024, 157014, 700392, 1849980, 2177010, 479532

Offset: 0

Gary W. Adamson, Sep 16 2008

Right border = A144251: (1, 1, 2, 6, 24, 122, 758,...) with row sums = the same sequence shifted. Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
  1;
  1, 1;
  1, 3, 2;
  1, 5, 12, 6;
  1, 7, 30, 60, 24;
  1, 9, 56, 210, 360, 122;
  1, 11, 90, 504, 1680, 2562, 758;
  1, 13, 132, 990, 5040, 15372, 21224, 5606;
  ...
The triangle is generated from A054142 and its own eigensequence, A144251.
Triangle A054142 =
  1;
  1, 1;
  1, 3, 1;
  1, 5, 6, 1;
  1, 7, 15, 10, 1;
  ...
The eigensequence of A054142 = A144251: (1, 1, 2, 6, 24, 122, 758, 5606,...);
Example: row 3 of A144252 = (1, 5, 12, 6) = termwise products of (1, 5, 6, 1) and (1, 1, 2, 6) = (1*1, 5*1, 6*2, 1*6).

Cf. A144251, A054142, A125273, A085478

A054142(n, k) = binomial(2*n-k, k);
V144251(nn) = my(v=vector(nn)); v[1] = 1; for (n=2, nn, v[n] = sum(k=0, n-1, A054142(n-2,k)*v[k+1]);); v;
row(n) = my(v=V144251(n+1)); vector(n+1, k, A054142(n,k-1) * v[k]); \\ Michel Marcus, Jan 18 2025

Eigentriangle by rows, T(n,k) = A054142(n,k) * A144251(k); were A144251 = the eigensequence of triangle A054142.

More terms from Michel Marcus, Jan 18 2025

A166697 A "Morgan Voyce" transform of A103210.

Original entry on oeis.org

1, 4, 25, 187, 1552, 13771, 127927, 1228576, 12099751, 121538581, 1240336660, 12824049277, 134043231781, 1414108869268, 15037450664317, 161014687970191, 1734550886346592, 18785969304551263, 204432608804093155

Offset: 0

Paul Barry, Oct 18 2009

Partial sums of A166696.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A085478, A103210, A166696

A166697 := proc(n)
    add(A166696(k),k=0..n) ;
end proc: # R. J. Mathar, Feb 10 2015

CoefficientList[Series[(1 - 3*t + t^2 - Sqrt[1 - 14*t + 27*t^2 - 14*t^3 + t^4])/(4*t*(1 - t)), {t, 0, 50}], t] (* G. C. Greubel, May 23 2016 *)

G.f.: (1-3x+x^2-sqrt(1-14x+27x^2-14x^3+x^4))/(4x(1-x));
G.f.: 1/(1-x-3x/(1-x-2x/(1-x-3x/(1-x-2x/(1-x-3x/(1-x-2x/(1-.... (continued fraction);
a(n) = Sum_{k=0..n} C(n+k,2k)*A103210(k).
a(n) = Sum_{k=0..n} A085478(n,k)*A103210(k). - Philippe Deléham, Nov 16 2013
Conjecture: (n+1)*a(n) + 3*(-5*n+2)*a(n-1) + (41*n-61)*a(n-2) + (-41*n+103)*a(n-3) + 3*(5*n-18)*a(n-4) + (-n+5)*a(n-5) = 0. - R. J. Mathar, Feb 10 2015

A171822 Triangle T(n,k) = binomial(2n-k, k)binomial(n+k, 2*k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 30, 30, 1, 1, 70, 225, 70, 1, 1, 135, 980, 980, 135, 1, 1, 231, 3150, 7056, 3150, 231, 1, 1, 364, 8316, 34650, 34650, 8316, 364, 1, 1, 540, 19110, 132132, 245025, 132132, 19110, 540, 1, 1, 765, 39600, 420420, 1288287, 1288287, 420420, 39600, 765, 1

Offset: 0

Roger L. Bagula, Dec 19 2009

			Triangle begins as:
  1;
  1,    1;
  1,    9,     1;
  1,   30,    30,       1;
  1,   70,   225,      70,       1;
  1,  135,   980,     980,     135,       1;
  1,  231,  3150,    7056,    3150,     231,       1;
  1,  364,  8316,   34650,   34650,    8316,     364,       1;
  1,  540, 19110,  132132,  245025,  132132,   19110,     540,     1;
  1,  765, 39600,  420420, 1288287, 1288287,  420420,   39600,   765,    1;
  1, 1045, 75735, 1166880, 5465460, 9018009, 5465460, 1166880, 75735, 1045, 1;

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

Cf. A054142, A085478, A183160.

[Binomial(2*n-k, k)*Binomial(n+k, 2*k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 22 2021

Table[Binomial[2*n-k, k]*Binomial[n+k, 2*k], {n,0,10}, {k,0,n}]//Flatten

flatten([[binomial(2*n-k, k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 22 2021

T(n, k) = binomial(2*n-k, k)*binomial(n+k, 2*k) = A054142(n, k)*A085478(n, k).

Sum_{k=0..n} T(n, k) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1) = A183160(n). - G. C. Greubel, Feb 22 2021

Edited by G. C. Greubel, Feb 22 2021

cp's OEIS Frontend

A114164 Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).

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A123878 Product of signed and unsigned Morgan-Voyce triangles.

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A123970 Triangle read by rows: T(0,0)=1; T(n,k) is the coefficient of x^(n-k) in the monic characteristic polynomial of the n X n matrix (min(i,j)) (i,j=1,2,...,n) (0 <= k <= n, n >= 1).

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A190909 Triangle read by rows: T(n,k) = binomial(n+k,n-k) * k! / floor(k/2)!^2.

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A108367 L(n,-n), where L is defined as in A108299.

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A114192 Riordan array (1/(1-2x),x/(1-2x)^2).

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A144251 Eigensequence of triangle A054142.

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A171822 Triangle T(n,k) = binomial(2n-k, k)binomial(n+k, 2*k), read by rows.