A098158 -id:A098158 - cp's OEIS frontend

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

Original entry on oeis.org

1, 5, 27, 155, 929, 5725, 35883, 227155, 1446241, 9237845, 59114907, 378678635, 2427143489, 15561826285, 99793962603, 640017621475, 4104915074881, 26328745454885, 168874407826587, 1083182932803515, 6947717948023649

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083879.

Inverse binomial transform of A147957. 5th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A083878, A006012.

[ n eq 1 select 1 else n eq 2 select 5 else 10*Self(n-1)-23*Self(n-2): n in [1..21] ]; // Klaus Brockhaus, Dec 16 2008

LinearRecurrence[{10,-23},{1,5},30] (* Harvey P. Dale, May 14 2018 *)

a(n)=if(n<0,0,polsym(23-10*x+x^2,n)[n+1]/2)

G.f.: (1-5x)/(1-10x+23x^2).
E.g.f.: exp(5x)cosh(x*sqrt(2)).
a(n) = ((5-sqrt(2))^n + (5+sqrt(2))^n)/2;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)*2^k.
a(n) = (Sum_{k=0..n} A098158(n,k)*5^(2k)*2^(n-k))/5^n. - Philippe Deléham, Nov 30 2008

Typo in definition corrected by Klaus Brockhaus, Dec 16 2008

A104537 Expansion of g.f.: (1+x)/(1+2*x+4x^2).

Original entry on oeis.org

1, -1, -2, 8, -8, -16, 64, -64, -128, 512, -512, -1024, 4096, -4096, -8192, 32768, -32768, -65536, 262144, -262144, -524288, 2097152, -2097152, -4194304, 16777216, -16777216, -33554432, 134217728, -134217728, -268435456, 1073741824, -1073741824, -2147483648, 8589934592

Offset: 0

Paul Barry, Mar 13 2005

a(n+1) is the Hankel transform of C(2n,n)-C(2n+2,n+1). - Paul Barry, Mar 14 2008

a(n+1) is the Hankel transform of C(2n,n)-2*C(n)=((n-1)/(n+1))*C(2n,n), where C(n)=A000108(n). - Paul Barry, Mar 14 2008

Index entries for linear recurrences with constant coefficients, signature (-2,-4).

Cf. A000108, A098158.

I:=[1, -1]; [n le 2 select I[n] else -2*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 16 2014

A104537:=n->2^n*cos(2*Pi*n/3): seq(A104537(n), n=0..40); # Wesley Ivan Hurt, Nov 16 2014

CoefficientList[Series[(1 + x) / (1 + 2 x + 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 16 2014 *)
LinearRecurrence[{-2,-4},{1,-1},40] (* Harvey P. Dale, Dec 02 2019 *)

G.f.: (1+x)/(1+2*x+4x^2).
a(n) = -2*a(n-1) - 4*a(n-2).
a(n) = 2^n*cos(2*Pi*n/3).
a(n) = Sum_{k=0..n} A098158(n,k)*(-1)^k*3^(n-k). - Philippe Deléham, Nov 14 2008
a(n) = (3^n/2^n)*Product_{i=1..n} (1/3 - tan((i-1/2)*Pi/(2*n))^2). - Gerry Martens, May 26 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = b(n) + b(n-1) where b(n) = 2^n*A049347(n). - R. J. Mathar, May 21 2019
Sum_{n>=0} 1/a(n) = -4/7. - Amiram Eldar, Feb 14 2023

A117411 Skew triangle associated to the Euler numbers.

Original entry on oeis.org

1, 0, 1, 0, -4, 1, 0, 0, -12, 1, 0, 0, 16, -24, 1, 0, 0, 0, 80, -40, 1, 0, 0, 0, -64, 240, -60, 1, 0, 0, 0, 0, -448, 560, -84, 1, 0, 0, 0, 0, 256, -1792, 1120, -112, 1, 0, 0, 0, 0, 0, 2304, -5376, 2016, -144, 1, 0, 0, 0, 0, 0, -1024, 11520, -13440, 3360, -180, 1, 0, 0, 0, 0, 0, 0, -11264, 42240, -29568, 5280, -220, 1

Offset: 0

Paul Barry, Mar 13 2006

Inverse is A117414. Row sums of the inverse are the Euler numbers A000364.

Triangle, read by rows, given by [0,-4,4,0,0,0,0,0,0,0,...] DELTA [1,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2009

			Triangle begins
  1;
  0,  1;
  0, -4,   1;
  0,  0, -12,   1;
  0,  0,  16, -24,    1;
  0,  0,   0,  80,  -40,     1;
  0,  0,   0, -64,  240,   -60,      1;
  0,  0,   0,   0, -448,   560,    -84,      1;
  0,  0,   0,   0,  256, -1792,   1120,   -112,      1;
  0,  0,   0,   0,    0,  2304,  -5376,   2016,   -144,      1;
  0,  0,   0,   0,    0, -1024,  11520, -13440,   3360,   -180,    1;
  0,  0,   0,   0,    0,     0, -11264,  42240, -29568,   5280, -220,    1;
  0,  0,   0,   0,    0,     0,   4096, -67584, 126720, -59136, 7920, -264, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000364, A006495 (row sums), A098158, A117413, A117414.

Cf. A000217, A000332, A000579, A000581, A262710.

A117411:= func< n,k | (-4)^(n-k)*(&+[Binomial(n,k-j)*Binomial(j,n-k): j in [0..n-k]]) >;
[A117411(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 07 2022

T[n_,k_]:= T[n,k]= (-4)^(n-k)*Sum[Binomial[n, k-j]*Binomial[j, n-k], {j,0,n-k}];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 07 2022 *)

def A117411(n,k): return (-4)^(n-k)*sum(binomial(n,k-j)*binomial(j,n-k) for j in (0..n-k))
flatten([[A117411(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Sep 07 2022

Sum_{k=0..n} T(n, k) = A006495(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A117413(n).
T(n, k) = (-4)^(n-k)*Sum_{j=0..n-k} C(n,k-j)*C(j,n-k).
G.f.: (1-x*y)/(1-2x*y+x^2*y(y+4)). - Paul Barry, Mar 14 2006
T(n, k) = (-4)^(n-k)*A098158(n,k). - Philippe Deléham, Nov 01 2009
T(n, k) = 2*T(n-1,k-1) - 4*T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 31 2013
From G. C. Greubel, Sep 07 2022: (Start)
T(n, n) = 1.
T(n, n-1) = -4*A000217(n-1), n >= 1.
T(n, n-2) = (-4)^2 * A000332(n), n >= 2.
T(n, n-3) = (-4)^3 * A000579(n), n >= 3.
T(n, n-4) = (-4)^4 * A000581(n), n >= 4.
T(2*n, n) = A262710(n). (End)

A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 6, 38, 252, 1732, 12216, 87704, 637104, 4663312, 34298208, 253025888, 1870171584, 13839178816, 102484311936, 759279663488, 5626889356032, 41707163713792, 309171726460416, 2292017151256064, 16992367115418624, 125979822242317312, 934017384983574528, 6924894663564105728

Offset: 0

Al Hakanson (hawkuu(AT)blogspot.com), Nov 17 2008

6th binomial transform of A077957. Binomial transform of A083880. Inverse binomial transform of A147958. - Philippe Deléham, Nov 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-34).

Cf. A077957, A083880, A098158, A147958.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((6+r2)^n+(6-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{12, -34}, {1, 6}, 50] (* G. C. Greubel, Aug 17 2018 *)

my(x='x+O('x^50)); Vec((1-6*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 12*a(n-1) - 34*a(n-2), n > 1; a(0)=1, a(1)=6.
G.f.: (1 - 6*x)/(1 - 12*x + 34*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*2^(n-k))/6^n. (End)
E.g.f.: exp(6*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A147958 a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 7, 51, 385, 2993, 23807, 192627, 1577849, 13036417, 108350935, 904201491, 7566326929, 63431106929, 532418131343, 4472591813139, 37592633210825, 316085049734017, 2658336935367463, 22360719757645683, 188108240644768801

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

nonn

7th binomial transform of A077957. Binomial transform of A147957. Inverse binomial transform of A147959. - Philippe Deléham, Nov 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14, -47).

Cf. A077957, A098158, A147957, A147959.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((7+r2)^n+(7-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{14, -47}, {1, 7}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-7*x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 14*a(n-1) - 47*a(n-2), n > 1; a(0)=1, a(1)=7.
G.f.: (1 - 7*x)/(1 - 14*x + 47*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2k)*2^(n-k))/7^n. (End)
E.g.f.: exp(7*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A147959 a(n) = ((8 + sqrt(2))^n + (8 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 8, 66, 560, 4868, 43168, 388872, 3545536, 32618512, 302072960, 2810819616, 26244590336, 245642629184, 2303117466112, 21620036448384, 203127300275200, 1909594544603392, 17959620096591872, 168959059780059648

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

Binomial transform of A147958. Inverse binomial transform of A147960. 8th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

			a(3) = ((8 + sqrt(2))^3 + (8 - sqrt(2))^3)/2
     = (8^3 + 3*8^2*sqrt(2) + 3*8*2 + 2*sqrt(2)
      + 8^3 - 3*8^2*sqrt(2) + 3*8*2 - 2*sqrt(2))/2
     =  8^3                 + 3*8*2
     =  560.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-62).

Cf. A077957, A147958, A147960.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((8+r2)^n+(8-r2)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{16, -62}, {1, 8}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-8*x)/(1-16*x+62*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 16*a(n-1) - 62*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1 - 8*x)/(1 - 16*x + 62*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*8^(2k)*2^(n-k))/8^n. (End)
E.g.f.: exp(8*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A147960 a(n) = ((9 + sqrt(2))^n + (9 - sqrt(2))^n)/2.

Original entry on oeis.org

1, 9, 83, 783, 7537, 73809, 733139, 7365591, 74662657, 762046137, 7818480563, 80531005311, 831898131121, 8612216940609, 89299952572403, 927034007995143, 9631915890692737, 100138799400852969, 1041577033850627219

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

nonn

Binomial transform of A147959. 9th binomial transform of A077957. - Philippe Deléham, Nov 30 2008

Hankel transform is := [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

G. C. Greubel, Table of n, a(n) for n = 0..980
Index entries for linear recurrences with constant coefficients, signature (18, -79).

Cf. A077957, A098158, A147959.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((9+r2)^n+(9-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{18, -79}, {1, 9}, 50] (* G. C. Greubel, Aug 17 2018 *)

x='x+O('x^30); Vec((1-9*x)/(1-18*x+79*x^2)) \\ G. C. Greubel, Aug 17 2018

From Philippe Deléham, Nov 19 2008: (Start)
a(n) = 18*a(n-1) - 79*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1 - 9*x)/(1 - 18*x + 79*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*9^(2k)*2^(n-k))/9^n. (End)
E.g.f.: exp(9*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008

A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.

Original entry on oeis.org

1, 4, 18, 88, 452, 2384, 12744, 68576, 370192, 2001472, 10829088, 58612096, 317289536, 1717746944, 9299922048, 50350919168, 272608444672, 1475954689024, 7991119286784, 43265588647936, 234249039168512, 1268274072276992

Offset: 0

Paul Barry, May 08 2003

Binomial transform of A083878.
4th binomial transform of A077957. Inverse binomial transform of A083880. - Philippe Deléham, Nov 30 2008
From L. Edson Jeffery, Apr 26 2011: (Start)
Let G be the Gram matrix
  G =
  (4  1  0  1)
  (1  4  1  0)
  (0  1  4 -1)
  (1  0 -1  4)
of A028997. Then a(n) = (1/4)*Trace(G^n). (End)

Index entries for linear recurrences with constant coefficients, signature (8,-14).

Cf. A028997, A083880.

LinearRecurrence[{8,-14},{1,4},30] (* Harvey P. Dale, May 08 2013 *)

a(n) = 2^((n-2)/2)*(2*sqrt(2)-1)^n + 2^((n-2)/2)*(2*sqrt(2)+1)^n;
a(n) = Sum_{k=0..n} C(n, 2k)*5^(n-2k)2^k.
G.f.: (1-4x)/(1-8x+14x^2).
E.g.f.: exp(4x)cosh(x*sqrt(2)).
((4+sqrt(2))^n + (4-sqrt(2))^n)/2. Offset=0. a(3)=88. - Al Hakanson (hawkuu(AT)gmail.com), Oct 15 2008
a(n) = Sum_{k=0..n} A098158(n,k)*2^(3*k-n). - Philippe Deléham, Nov 30 2008

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A146963 a(n) = ((3 + sqrt(7))^n + (3 - sqrt(7))^n)/2.

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A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A083880 a(0)=1, a(1)=5, a(n) = 10*a(n-1) - 23*a(n-2), n >= 2.

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A104537 Expansion of g.f.: (1+x)/(1+2*x+4x^2).

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A117411 Skew triangle associated to the Euler numbers.

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A147957 a(n) = ((6 + sqrt(2))^n + (6 - sqrt(2))^n)/2.

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A147958 a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

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A083880 a(0)=1, a(1)=5, a(n) = 10a(n-1) - 23a(n-2), n >= 2.

A083879 a(0)=1, a(1)=4, a(n) = 8a(n-1) - 14a(n-2), n >= 2.