A027960 -id:A027960 - cp's OEIS frontend

A027969 a(n) = T(n, 2*n-7), T given by A027960.

3, 7, 18, 47, 120, 291, 661, 1404, 2801, 5283, 9484, 16305, 26990, 43215, 67191, 101782, 150639, 218351, 310614, 434419, 598260, 812363, 1088937, 1442448, 1889917, 2451243, 3149552, 4011573, 5068042, 6354135, 7909931, 9780906, 12018459, 14680471, 17831898, 21545399, 25902000, 30991795

Offset: 4

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

A column of triangle A027011.

List([4..50], n-> (90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040) # G. C. Greubel, Jul 01 2019

[(90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040: n in [4..50]]; // G. C. Greubel, Jul 01 2019

Table[(90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040, {n,4,50}] (* G. C. Greubel, Jul 01 2019 *)

for(n=4,50, print1((90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040, ", ")) \\ G. C. Greubel, Jul 01 2019

[(90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040 for n in (4..50)] # G. C. Greubel, Jul 01 2019

From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^4*(3-2x)*(1-x+x^2)*(1-4x+7x^2-4x^3+x^4)/(1-x)^8.
First differences of A027970. (End)
From G. C. Greubel, Jul 01 2019: (Start)
a(n) = (90720 -85548*n +38822*n^2 -10136*n^3 +1505*n^4 -77*n^5 -7*n^6 + n^7)/5040.
E.g.f.: (-90720 - 35280*x - 7560*x^2 - 1680*x^3 + (90720 - 55440*x + 17640*x^2 - 3360*x^3 + 630*x^4 - 42*x^5 + 14*x^6 + x^7)*exp(x))/5040. (End)

Terms a(35) onward added by G. C. Greubel, Jul 01 2019

A027970 a(n) = T(n, 2*n-8), T given by A027960.

Original entry on oeis.org

1, 4, 11, 29, 76, 196, 487, 1148, 2552, 5353, 10636, 20120, 36425, 63415, 106630, 173821, 275603, 426242, 644593, 955207, 1389626, 1987886, 2800249, 3889186, 5331634, 7221551, 9672794, 12822346, 16833919, 21901961, 28256096

Offset: 4

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

A column of triangle A026998.

List([4..40], n-> (-1169280 +1119312*n -517700*n^2 +148092*n^3 -26551*n^4 +2688*n^5 -70*n^6 -12*n^7 +n^8)/40320); # G. C. Greubel, Jul 01 2019

[(-1169280 +1119312*n -517700*n^2 +148092*n^3 -26551*n^4 +2688*n^5 -70*n^6 -12*n^7 +n^8)/40320: n in [4..40]]; // G. C. Greubel, Jul 01 2019

Table[(-1169280 +1119312*n -517700*n^2 +148092*n^3 -26551*n^4 +2688*n^5 -70*n^6 -12*n^7 +n^8)/40320, {n, 4, 40}] (* G. C. Greubel, Jul 01 2019 *)

Vec(x^4*(x^8-5*x^7+11*x^6-10*x^5-x^4+10*x^3-11*x^2+5*x-1)/(x-1)^9 + O(x^40)) \\ Colin Barker, Nov 25 2014

for(n=4,40, print1((-1169280 +1119312*n -517700*n^2 +148092*n^3 -26551*n^4 +2688*n^5 -70*n^6 -12*n^7 +n^8)/40320, ", ")) \\ G. C. Greubel, Jul 01 2019

[(-1169280 +1119312*n -517700*n^2 +148092*n^3 -26551*n^4 +2688*n^5 -70*n^6 -12*n^7 +n^8)/40320 for n in (4..40)] # G. C. Greubel, Jul 01 2019

Sequence satisfies an 8-degree polynomial approximating A002878.
a(n) = (-1169280 +1119312*n -517700*n^2 +148092*n^3 -26551*n^4 +2688*n^5 -70*n^6 -12*n^7 +n^8)/40320. - Colin Barker, Nov 25 2014
G.f.: x^4*(x^8-5*x^7+11*x^6-10*x^5-x^4+10*x^3-11*x^2+5*x-1) / (x-1)^9. - Colin Barker, Nov 25 2014
From G. C. Greubel, Jul 01 2019: (Start)
a(n) = A027971(n+1) - A027971(n).
E.g.f.: (1169280 + 443520*x + 80640*x^2 + 6720*x^3 +(-1169280 +725760*x  -221760*x^2 +47040*x^3 -6720*x^4 +1008*x^5 -56*x^6 +16*x^7 +x^8)*exp(x) )/8!. (End)

A027971 T(n, 2n-9), T given by A027960.

Original entry on oeis.org

3, 7, 18, 47, 123, 319, 806, 1954, 4506, 9859, 20495, 40615, 77040, 140455, 247085, 420906, 696509, 1122751, 1767344, 2722551, 4112177, 6100063, 8900312, 12789498, 18121132, 25342683, 35015477, 47837823, 64671742

Offset: 5

Clark Kimberling

Colin Barker, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

A column of triangle A027011.

a:=[3,7,18,47, 123,319,806,1954,4506,9859];; for n in [11..40] do a[n]:=10*a[n-1]-45*a[n-2]+120*a[n-3]-210*a[n-4]+252*a[n-5]-210*a[n-6] +120*a[n-7] -45*a[n-8]+10*a[n-9]-a[n-10]; od; a; # G. C. Greubel, Sep 26 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^5*(3 -2*x)*(1-7*x+23*x^2-44*x^3+55*x^4-44*x^5+23*x^6-7*x^7+x^8)/(1-x)^10 )); // G. C. Greubel, Sep 26 2019

seq(coeff(series(x^5*(3-2*x)*(1 -7*x +23*x^2 -44*x^3 +55*x^4 -44*x^5 +23*x^6 -7*x^7 +x^8)/(1-x)^10, x, n+1), x, n), n = 5..40); # G. C. Greubel, Sep 26 2019

LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1}, {3,7,18,47, 123,319,806,1954,4506,9859},40] (* Harvey P. Dale, Aug 04 2017 *)

Vec(-x^5*(2*x-3)*(x^8-7*x^7+23*x^6-44*x^5+55*x^4-44*x^3+23*x^2 -7*x+1)/(x-1)^10 + O(x^40)) \\ Colin Barker, Nov 25 2014

def A027971_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^5*(3-2*x)*(1-7*x+23*x^2-44*x^3+55*x^4-44*x^5+23*x^6-7*x^7 +x^8)/(1-x)^10 ).list()
a=A027971_list(40); a[5:] # G. C. Greubel, Sep 26 2019

a(n) = (17055360 -16329024*n +7697736*n^2 -2299060*n^3 +462798*n^4 -60207*n^5 +4284*n^6 -30*n^7 -18*n^8 +n^9)/362880. - Colin Barker, Nov 25 2014

G.f.: x^5*(3-2*x)*(1 -7*x +23*x^2 -44*x^3 +55*x^4 -44*x^5 +23*x^6 -7*x^7 +x^8)/(1-x)^10. - Colin Barker, Nov 25 2014

A027972 T(n, 2n-10), T given by A027960.

Original entry on oeis.org

1, 4, 11, 29, 76, 199, 518, 1324, 3278, 7784, 17643, 38138, 78753, 155793, 296248, 543333, 964239, 1660748, 2783499, 4550843, 7273394, 11385571, 17485634, 26385946, 39175444, 57296576, 82639259, 117654736, 165492559

Offset: 5

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

A column of triangle A026998.

a:=[1, 4, 11, 29, 76, 199, 518, 1324, 3278, 7784, 17643];; for n in [12..40] do a[n]:=11*a[n-1]-55*a[n-2]+165*a[n-3]-330*a[n-4]+462*a[n-5] -462*a[n-6]+330*a[n-7]-165*a[n-8]+55*a[n-9]-11*a[n-10]+a[n-11]; od; a; # G. C. Greubel, Sep 26 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x^5*(1- 7*x+22*x^2-37*x^3+32*x^4+x^5-32*x^6+37*x^7-22*x^8+7*x^9 -x^10)/(1-x)^11 )); // G. C. Greubel, Sep 26 2019

seq(coeff(series(x^5*(1 -7*x +22*x^2 -37*x^3 +32*x^4 +x^5 -32*x^6 +37*x^7 -22*x^8 +7*x^9 -x^10)/(1-x)^11, x, n+1), x, n), n = 5..40); # G. C. Greubel, Sep 26 2019

Drop[CoefficientList[Series[x^5*(1 -7*x +22*x^2 -37*x^3 +32*x^4 +x^5 -32*x^6 +37*x^7 -22*x^8 +7*x^9 -x^10)/(1-x)^11, {x, 0, 40}], x], 5] (* G. C. Greubel, Sep 26 2019 *)

my(x='x+O('x^40)); Vec(x^5*(1 -7*x +22*x^2 -37*x^3 +32*x^4 +x^5 -32*x^6 +37*x^7 -22*x^8 +7*x^9 -x^10)/(1-x)^11) \\ G. C. Greubel, Sep 26 2019

def A027972_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^5*(1 -7*x +22*x^2 -37*x^3 +32*x^4 +x^5 -32*x^6 +37*x^7 -22*x^8 +7*x^9 -x^10)/(1-x)^11 ).list()
a=A027972_list(40); a[5:] # G. C. Greubel, Sep 26 2019

Sequence satisfies a 10-degree polynomial approximating A002878.
G.f.: x^5*(1 -7*x +22*x^2 -37*x^3 +32*x^4 +x^5 -32*x^6 +37*x^7 -22*x^8 +7*x^9 -x^10)/(1-x)^11. - R. J. Mathar, Jan 30 2011
a(n) = -76 +183941*n/2520 +386899*n^3/36288 -1747657*n^2/50400 -831241*n^4/362880 +11887*n^5/34560 -5807*n^6/172800 +41*n^7/24192 +n^8/60480 -n^9/145152 +n^10/3628800. - R. J. Mathar, Jan 30 2011

A027976 n-th diagonal sum of right justified array T given by A027960.

Original entry on oeis.org

1, 1, 4, 6, 10, 18, 29, 47, 78, 126, 204, 332, 537, 869, 1408, 2278, 3686, 5966, 9653, 15619, 25274, 40894, 66168, 107064, 173233, 280297, 453532, 733830, 1187362, 1921194, 3108557, 5029751, 8138310, 13168062, 21306372, 34474436, 55780809, 90255245, 146036056, 236291302, 382327358

Offset: 0

Clark Kimberling

nonn

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

Cf. A000032, A000045, A004695, A027960, A102283.

a:=[1,1,4,6,10];; for n in [6..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]-a[n-4]-a[n-5]; od; a; # G. C. Greubel, Sep 26 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1 + 2*x^2)/((1-x^3)*(1-x-x^2)) )); // G. C. Greubel, Sep 26 2019

seq(coeff(series((1 + 2*x^2)/((1-x^3)*(1-x-x^2)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Sep 26 2019

LinearRecurrence[{1,1,1,-1,-1}, {1,1,4,6,10}, 41] (* or *) Table[ (Fibonacci[n+1] +LucasL[n+2] -2*Sin[2*Pi*n/3]/Sqrt[3] -2)/2, {n,0,40}] (* G. C. Greubel, Sep 26 2019 *)

my(x='x+O('x^40)); Vec((1 + 2*x^2)/((1-x^3)*(1-x-x^2))) \\ G. C. Greubel, Sep 26 2019

def A027976_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1 + 2*x^2)/((1-x^3)*(1-x-x^2))).list()
A027976_list(40) # G. C. Greubel, Sep 26 2019

G.f.: (1 + 2*x^2)/((1-x^3)*(1-x-x^2)).
From G. C. Greubel, Sep 26 2019: (Start)
a(n) = (Fibonacci(n) + 4*Fibonacci(n+1) - A102283(n) - 2)/2.
a(n) = (Fibonacci(n+1) + Lucas(n+2) - 2*sin(2*Pi*n/3)/sqrt(3) - 2)/2. (End)

Terms a(28) onward added by G. C. Greubel, Sep 26 2019

A027981 a(n) = Sum_{k=0..2n} (k+1)T(n,k), T given by A027960.

Original entry on oeis.org

1, 10, 40, 124, 340, 868, 2116, 4996, 11524, 26116, 58372, 129028, 282628, 614404, 1327108, 2850820, 6094852, 12976132, 27525124, 58195972, 122683396, 257949700, 541065220, 1132462084, 2365587460, 4932501508, 10267656196, 21340618756, 44291850244, 91804925956, 190052302852, 392989507588, 811748818948, 1675037245444, 3453153705988

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A027960.

A027981:= func< n | 3*2^n*(2*n-1) + 4 >; // G. C. Greubel, Jun 07 2025

LinearRecurrence[{5,-8,4},{1,10,40},30] (* Harvey P. Dale, Apr 17 2015 *)

def A027981(n): return 3*2**n*(2*n-1) + 4 # G. C. Greubel, Jun 07 2025

a(2*n+1) = 6*(4*n+1) * 4^n + 4. - Ralf Stephan, Mar 22 2004
From R. J. Mathar, May 22 2013: (Start)
a(n) = 3*2^n*(2*n-1) + 4.
G.f.: (1 + 5*x - 2*x^2)/( (1-x)*(1-2*x)^2 ). (End)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3), with a(0)=1, a(1)=10, a(2)=40. - Harvey P. Dale, Apr 17 2015
E.g.f.: 3*(4*x - 1)*exp(2*x) + 4*exp(x). - Ilya Gutkovskiy, Apr 17 2016

More terms added by G. C. Greubel, Jun 07 2025

A027975 a(n) is the n-th diagonal sum of left justified array T given by A027960.

Original entry on oeis.org

1, 1, 4, 5, 8, 12, 16, 23, 31, 42, 57, 76, 102, 136, 181, 241, 320, 425, 564, 748, 992, 1315, 1743, 2310, 3061, 4056, 5374, 7120, 9433, 12497, 16556, 21933, 29056, 38492, 50992, 67551, 89487, 118546, 157041, 208036, 275590, 365080, 483629, 640673, 848712, 1124305, 1489388, 1973020

Offset: 0

Clark Kimberling

nonn

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

Cf. A027960.

a:=[1,1,4,5];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]-a[n-4]; od; a; # G. C. Greubel, Sep 26 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+2*x^2)/((1-x)*(1-x^2-x^3)) )); // G. C. Greubel, Sep 26 2019

seq(coeff(series((1+2*x^2)/((1-x)*(1-x^2-x^3)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Sep 26 2019

CoefficientList[Series[(1+2*x^2)/((1-x)*(1-x^2-x^3)), {x,0,40}], x] (* or *) LinearRecurrence[{1,1,0,-1}, {1,1,4,5}, 41] (* G. C. Greubel, Sep 26 2019 *)

my(x='x+O('x^40)); Vec((1+2*x^2)/((1-x)*(1-x^2-x^3))) \\ G. C. Greubel, Sep 26 2019

def A027975_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+2*x^2)/((1-x)*(1-x^2-x^3)) ).list()
A027975_list(40) # G. C. Greubel, Sep 26 2019

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

a(n) = a(n-2) + a(n-3) + 3. - Greg Dresden, May 18 2020

Terms a(32) onward added by G. C. Greubel, Sep 26 2019

A027977 a(n) = greatest number in row n of array T given by A027960.

Original entry on oeis.org

1, 3, 4, 8, 15, 28, 54, 101, 199, 373, 743, 1404, 2801, 5353, 10636, 20495, 40615, 78753, 155793, 303553, 599801, 1173183, 2316317, 4544731, 8968421, 17641499, 34801731, 68602923, 135308317, 267203186, 526966454, 1042217402, 2055373383, 4070330014, 8027429651, 15914813448, 31389204737, 62291326036, 122871494899

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A027960.

T[n_, k_]:= T[n, k]= If[k<0 || k>2*n, 0, If[k==0 || k==2*n, 1, If[k==1, 3, T[n-1, k-2] + T[n-1, k-1]]]]; (* T = A027960 *)
b[n_]:= b[n]= Table[T[n,k], {k,0,2*n}]//Union;
A027977[n_]:= Max[b[n]];
Table[A027977[n], {n,0,50}] (* G. C. Greubel, Jun 07 2025 *)

@CachedFunction
def T(n, k): # T = A027960
    if (k<0 or k>2*n): return 0
    elif (k==0 or k==2*n): return 1
    elif (k==1): return 3
    else: return T(n-1, k-2) + T(n-1, k-1)
def b(n): return sorted(set(flatten([T(n,k) for k in range(2*n+1)])))
def A027977(n): return max(b(n))
print([A027977(n) for n in range(51)]) # G. C. Greubel, Jun 07 2025

More terms added by G. C. Greubel, Jun 07 2025

A027978 a(n) = self-convolution of row n of array T given by A027960.

Original entry on oeis.org

1, 11, 42, 145, 473, 1484, 4529, 13543, 39870, 115937, 333781, 953056, 2702497, 7618115, 21365778, 59657329, 165926609, 459905588, 1270819025, 3501855007, 9625627686, 26398369601, 72248624077, 197361589960, 538199264833

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000032, A000045, A027960.

List([0..40], n-> 2*(n+1)*Lucas(1,-1,2*n)[2] + Fibonacci(2*n-4) ); # G. C. Greubel, Oct 01 2019

[2*(n+1)*Lucas(2*n) + Fibonacci(2*n-4): n in [0..40]]; // G. C. Greubel, Oct 01 2019

with(combinat); f:=fibonacci; seq(2*(n+1)*(f(2*n+1) + f(2*n-1)) + f(2*n-4), n=0..40); # G. C. Greubel, Oct 01 2019

Table[2*(n+1)*LucasL[2*n] + Fibonacci[2*n-4], {n, 0, 40}] (* G. C. Greubel, Oct 01 2019 *)

vector(41, n, f=fibonacci; 2*n*(f(2*n-1) + f(2*n-3)) + f(2*n-6)) \\ G. C. Greubel, Oct 01 2019

[2*(n+1)*lucas_number2(2*n,1,-1) + fibonacci(2*n-4) for n in (0..40)] # G. C. Greubel, Oct 01 2019

From Colin Barker, Feb 25 2015: (Start)
a(n) = 5*a(n-1) - 5*a(n-2) - 5*a(n-3) + 5*a(n-4) - a(n-5).
G.f.: (1 +5*x -13*x^2 +8*x^3)/(1-3*x+x^2)^2. (End)
a(n) = 2*(n+1)*Lucas(2*n) + Fibonacci(2*n-4). - G. C. Greubel, Oct 01 2019

A027979 a(n) = Sum_{k=0..n} T(n,k)*T(n,2n-k), T given by A027960.

Original entry on oeis.org

1, 10, 29, 97, 297, 904, 2685, 7876, 22823, 65533, 186691, 528370, 1486969, 4164382, 11613137, 32264089, 89339325, 246645436, 679111413, 1865340568, 5112351131, 13983383605, 38177371159, 104055773542, 283171508977

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,-5,5,-1).

Cf. A000032, A000045, A027960.

List([0..40], n-> (n+1)*Lucas(1,-1,2*n)[2] + 3*Fibonacci(2*n) -(-1)^n ); # G. C. Greubel, Oct 01 2019

[(n+1)*Lucas(2*n) + 3*Fibonacci(2*n) -(-1)^n: n in [0..40]]; // G. C. Greubel, Oct 01 2019

f:= combinat[fibonacci]: seq((n+1)*(f(2*n+1) + f(2*n-1)) + 3*f(2*n) -(-1)^n, n=0..40); # G. C. Greubel, Oct 01 2019

Table[(n+1)*LucasL[2*n] +3*Fibonacci[2*n] -(-1)^n, {n,0,40}] (* G. C. Greubel, Oct 01 2019 *)

vector(41, n, f=fibonacci; n*(f(2*n-1) + f(2*n-3)) + 3*f(2*n-2) +(-1)^n) \\ G. C. Greubel, Oct 01 2019

[(n+1)*lucas_number2(2*n,1,-1) + 3*fibonacci(2*n) -(-1)^n for n in (0..40)] # G. C. Greubel, Oct 01 2019

G.f.: (1 +5*x -16*x^2 +7*x^3 +2*x^4)/((1+x)*(1-3*x+x^2)^2). - Colin Barker, Nov 25 2014

a(n) = (n+1)*Lucas(2*n) + 3*Fibonacci(2*n) - (-1)^n. - G. C. Greubel, Oct 01 2019

cp's OEIS Frontend

A027969 a(n) = T(n, 2*n-7), T given by A027960.

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A027970 a(n) = T(n, 2*n-8), T given by A027960.

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Magma

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PARI

Sage

Formula

A027971 T(n, 2n-9), T given by A027960.

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GAP

Magma

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PARI

Sage

Formula

A027972 T(n, 2n-10), T given by A027960.

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GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A027976 n-th diagonal sum of right justified array T given by A027960.

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GAP

Magma

Maple

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Sage

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Extensions

A027981 a(n) = Sum_{k=0..2*n} (k+1)*T(n,k), T given by A027960.

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Magma

A027981 a(n) = Sum_{k=0..2n} (k+1)T(n,k), T given by A027960.