A154996 -id:A154996 - cp's OEIS frontend

A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 10, 6, 1, 0, 8, 22, 21, 8, 1, 0, 13, 45, 59, 36, 10, 1, 0, 21, 88, 147, 124, 55, 12, 1, 0, 34, 167, 339, 366, 225, 78, 14, 1, 0, 55, 310, 741, 976, 770, 370, 105, 16, 1, 0, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 0, 144, 1020, 3174, 5696, 6505, 4920, 2485, 824, 171, 20, 1

Offset: 0

Philippe Deléham, Jan 20 2009

A Fibonacci convolution triangle; Riordan array (1, x*(1+x)/(1-x-x^2)).

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  4,  1;
  0,  5, 10,  6,  1;
  0,  8, 22, 21,  8,  1;
  0, 13, 45, 59, 36, 10, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
Cf. A000012, A155020, A154964, A154968, A154996, A154997, A154999, A155000, A155001, A155017.
Cf. A000045, A154929.
T(2n,n) gives A262910.
Cf. A291385.

T:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n-j,j)*Binomial(n-j,k)*k/(n-j): j in [0..Floor(n/2)]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 26 2021

# Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n+1)); # Peter Luschny, Oct 19 2022

T[n_, k_]:= If[n==0, 1, Sum[Binomial[n-j, j]*Binomial[n-j, k]*k/(n-j), {j, 0, Floor[n/2]}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 26 2021 *)

def T(n,k): return 1 if n==0 else sum( binomial(n-j,j)*binomial(n-j,k)*k/(n-j) for j in (0..n//2) )
flatten([[T(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 26 2021

Recurrence: T(n+2,k+1) = T(n+1,k+1) + T(n+1,k) + T(n,k+1) + T(n,k).
Explicit formula: T(n,k) = Sum_{i=0..floor(n/2)} binomial(n-i, i)*binomial(n-i, k)*k/(n-i), for n > 0.
G.f.: (1-x-x^2)/(1-(1+y)*x-(1+y)*x^2). - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A155020(n), A154964(n), A154968(n), A154996(n), A154997(n), A154999(n), A155000(n), A155001(n), A155017(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n, k)*(m-1)^k = (1/m)*[n=0] - (m-1)*(i*sqrt(m))^(n-2)*ChebyshevU(n, -i*sqrt(m)/2). - G. C. Greubel, Mar 26 2021
Sum_{k=0..n} k * T(n,k) = A291385(n-1) for n>=1. - Alois P. Heinz, Sep 29 2022

Typos in two terms corrected by Alois P. Heinz, Aug 08 2015

A154997 a(n) = 6a(n-1) + 30a(n-2), n>2; a(0)=1, a(1)=1, a(2)=11.

Original entry on oeis.org

1, 1, 11, 96, 906, 8316, 77076, 711936, 6583896, 60861456, 562685616, 5201957376, 48092312736, 444612597696, 4110444968256, 38001047740416, 351319635490176, 3247949245153536, 30027284535626496, 277602184568365056

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,30).

Cf. A154996, A154999, A155000, A155001.

I:=[1,11]; [1] cat [n le 2 select I[n] else 6*(Self(n-1) +5*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 21 2021

m:=30; S:=series( (1-5*x-25*x^2)/(1-6*x-30*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 21 2021

Join[{1},LinearRecurrence[{6,30},{1,11},20]] (* Harvey P. Dale, Feb 07 2012 *)

def A154996_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-5*x-25*x^2)/(1-6*x-30*x^2) ).list()
A154996_list(30) # G. C. Greubel, Apr 21 2021

G.f.: (1 -5*x -25*x^2)/(1 -6*x -30*x^2).

a(n+1) = Sum_{k=0..n} A154929(n,k)*5^(n-k).

A154999 a(n) = 7a(n-1) + 42a(n-2), n>2; a(0)=1, a(1)=1, a(2)=13.

Original entry on oeis.org

1, 1, 13, 133, 1477, 15925, 173509, 1883413, 20471269, 222402229, 2416608901, 26257155925, 285297665317, 3099884206069, 33681691385797, 365966976355477, 3976399872691813, 43205412115772725, 469446679463465221

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (7,42).

Cf. A154929, A154996, A154997, A155000, A155001.

I:=[1,13]; [1] cat [n le 2 select I[n] else 7*(Self(n-1) +6*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

LinearRecurrence[{7,42}, {1,1,13}, 31] (* G. C. Greubel, Apr 20 2021 *)
CoefficientList[Series[(1-6x-36x^2)/(1-7x-42x^2),{x,0,20}],x] (* Harvey P. Dale, Jan 14 2022 *)

def A154999_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-6*x-36*x^2)/(1-7*x-42*x^2) ).list()
A154999_list(30) # G. C. Greubel, Apr 20 2021

a(n+1) = Sum_{k=0..n} A154929(n,k)*6^(n-k).

G.f.: (1 - 6*x - 36*x^2)/(1 - 7*x - 42*x^2). - G. C. Greubel, Apr 20 2021

More terms from Philippe Deléham, Jan 27 2009

Corrected by D. S. McNeil, Aug 20 2010

A155000 a(n) = 8a(n-1) + 56a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=15.

Original entry on oeis.org

1, 1, 15, 176, 2248, 27840, 348608, 4347904, 54305280, 677924864, 8464494592, 105679749120, 1319449690112, 16473663471616, 205678490419200, 2567953077764096, 32061620085587968, 400298333039493120

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (8,56).

Cf. A154996, A154997, A154999, A155001, A155112.

I:=[1,15]; [1] cat [n le 2 select I[n] else 8*(Self(n-1) +7*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

m:=30; S:=series( (1-7*x-49*x^2)/(1-8*x-56*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 20 2021

Join[{1},LinearRecurrence[{8,56},{1,15},20]] (* Harvey P. Dale, Dec 11 2012 *)

def A155000_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-7*x-49*x^2)/(1-8*x-56*x^2) ).list()
A155000_list(30) # G. C. Greubel, Apr 20 2021

a(n) = Sum_{k=0..n} A155112(n,k)*7^(n-k). - Philippe Deléham, Jan 27 2009

G.f.: 1 + x*(1+7*x)/(1-8*x-56*x^2). - Harvey P. Dale, Dec 11 2012

More terms from Philippe Deléham, Jan 27 2009

A155001 a(n) = 9a(n-1) + 72a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=17.

Original entry on oeis.org

1, 1, 17, 225, 3249, 45441, 642897, 9057825, 127809009, 1802444481, 25424248977, 358594243425, 5057894117169, 71339832581121, 1006226869666257, 14192509772837025, 200180922571503729, 2823489006787799361

Offset: 0

Philippe Deléham, Jan 18 2009

nonn

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

G. C. Greubel, Table of n, a(n) for n = 0..850
Index entries for linear recurrences with constant coefficients, signature (9,72).

Cf. A154929, A154996, A154997, A154999, A155000.

I:=[1,17]; [1] cat [n le 2 select I[n] else 9*(Self(n-1) +8*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

a[0] := 1: a[1] := 1: a[2] := 17: for n from 3 to 25 do a[n] := 9*a[n-1]+72*a[n-2] end do: seq(a[n], n = 0 .. 17); # Emeric Deutsch, Jan 21 2009

LinearRecurrence[{9,72},{1,1,17},20] (* Harvey P. Dale, Apr 26 2016 *)

def A155001_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-8*x-64*x^2)/(1-9*x-72*x^2) ).list()
A155001_list(30) # G. C. Greubel, Apr 20 2021

a(n+1) = Sum_{k=0..n} A154929(n,k)*8^(n-k).

G.f.: (1 - 8*x - 64*x^2)/(1 - 9*x - 72*x^2). - G. C. Greubel, Apr 20 2021

Corrected by Philippe Deléham, Jan 21 2009

Corrected and extended by Emeric Deutsch and R. J. Mathar, Jan 21 2009

cp's OEIS Frontend

A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A154997 a(n) = 6*a(n-1) + 30*a(n-2), n>2; a(0)=1, a(1)=1, a(2)=11.

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A154999 a(n) = 7*a(n-1) + 42*a(n-2), n>2; a(0)=1, a(1)=1, a(2)=13.

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Magma

Mathematica

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A155000 a(n) = 8*a(n-1) + 56*a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=15.

Views

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Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A155001 a(n) = 9*a(n-1) + 72*a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=17.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A154997 a(n) = 6a(n-1) + 30a(n-2), n>2; a(0)=1, a(1)=1, a(2)=11.

A154999 a(n) = 7a(n-1) + 42a(n-2), n>2; a(0)=1, a(1)=1, a(2)=13.

A155000 a(n) = 8a(n-1) + 56a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=15.

A155001 a(n) = 9a(n-1) + 72a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=17.