A160747 -id:A160747 - cp's OEIS frontend

A160837 G.f.: (1+38x+262x^2+475x^3+254x^4+37*x^5+x^6)/(1-x)^7.

1, 45, 556, 3457, 14317, 45565, 120772, 280001, 586225, 1132813, 2052084, 3524929, 5791501, 9162973, 14034364, 20898433, 30360641, 43155181, 60162076, 82425345, 111172237, 147833533, 194064916, 251769409, 323120881, 410588621, 516962980

Offset: 0

N. J. A. Sloane, Nov 18 2009

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[1+ n*(n+1)*(89*n^4+183*n^3+427*n^2+333*n+288)/60: n in [0..30]]; // Vincenzo Librandi, Sep 19 2011

CoefficientList[Series[(1+38x+262x^2+475x^3+254x^4+37x^5+x^6)/(1-x)^7, {x,0,40}], x] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1, 45, 556,3457,14317,45565,120772}, 40] (* Harvey P. Dale, Nov 27 2016 *)

x='x+O('x^30); Vec((1+38*x+262*x^2+475*x^3+254*x^4+37*x^5+x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

From R. J. Mathar, Dec 16 2009: (Start)
a(n) = 1+24/5*n+38/3*n^3+207/20*n^2+61/6*n^4+68/15*n^5+89/60*n^6.
a(n) = 1+ n*(n+1)*(89*n^4+183*n^3+427*n^2+333*n+288)/60. (End)

A160838 G.f.: (1+38x+263x^2+484x^3+263x^4+38*x^5+x^6)/(1-x)^7.

Original entry on oeis.org

1, 45, 557, 3473, 14417, 45965, 121997, 283137, 593281, 1147213, 2079309, 3573329, 5873297, 9295469, 14241389, 21212033, 30823041, 43821037, 61101037, 83724945, 112941137, 150205133, 197201357, 255865985, 328410881, 417348621, 525518605

Offset: 0

N. J. A. Sloane, Nov 18 2009

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[68*n^6/45 +68*n^5/15 +91*n^4/9 +38*n^3/3 +467*n^2/45 +24*n/5 +1: n in [0..30]]; // Vincenzo Librandi, Sep 19 2011

CoefficientList[Series[(1+38x+263x^2+484x^3+263x^4+38x^5+x^6)/(1-x)^7, {x,0,30}], x] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,45, 557, 3473,14417,45965,121997}, 30] (* Harvey P. Dale, Sep 17 2011 *)

x='x+O('x^30); Vec((1+38*x+263*x^2+484*x^3+263*x^4+38*x^5+x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

a(n) = 68*n^6/45 +68*n^5/15 +91*n^4/9 +38*n^3/3 +467*n^2/45 +24*n/5 +1. - R. J. Mathar, Sep 11 2011

a(0)=1, a(1)=45, a(2)=557, a(3)=3473, a(4)=14417, a(5)=45965, a(6)=121997, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)- 7*a(n-6)+a(n-7). - Harvey P. Dale, Sep 17 2011

A160839 Expansion of (78+1116x+3492x^2+3237x^3+927x^4+72*x^5+x^6)/(1-x)^7.

Original entry on oeis.org

78, 1662, 13488, 65481, 231486, 660921, 1619353, 3537997, 7072138, 13168476, 23141394, 38758149, 62332986, 96830175, 145975971, 214379497, 307662550, 432598330, 597259092, 811172721, 1085488230, 1433150181, 1869082029

Offset: 0

N. J. A. Sloane, Nov 18 2009

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[8923*n^6/720 +18691*n^5/240 +35375*n^4/144 +7219*n^3/16 +178361*n^2/360 +9043*n/30 + 78: n in [0..30]]; // Vincenzo Librandi, Sep 19 2011

seq(coeff(series((78+1116*x+3492*x^2+3237*x^3+927*x^4+72*x^5+x^6)/(1-x)^7, x,n+1),x,n),n=0..25); # Muniru A Asiru, Apr 29 2018

Table[8923*n^6/720 +18691*n^5/240 +35375*n^4/144 +7219*n^3/16 +178361*n^2/360 +9043*n/30 + 78, {n, 0, 30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1}, {78, 1662, 13488, 65481, 231486, 660921, 1619353}, 30] (* G. C. Greubel, Apr 28 2018 *)

x='x+O('x^30); Vec((78+1116*x+3492*x^2+3237*x^3+927*x^4 +72*x^5 +x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

a(n) = 8923*n^6/720 +18691*n^5/240 +35375*n^4/144 +7219*n^3/16 +178361*n^2/360 +9043*n/30 + 78. - R. J. Mathar, Sep 11 2011

A160840 Expansion of (1+147x+1230x^2+1885x^3+714x^4+63*x^5+x^6)/(1-x)^7.

Original entry on oeis.org

1, 154, 2287, 14695, 60907, 192493, 505912, 1163401, 2417905, 4642048, 8361145, 14290255, 23375275, 36838075, 56225674, 83463457, 120912433, 171430534, 238437955, 325986535, 438833179, 582517321, 763442428, 988961545, 1267466881

Offset: 0

N. J. A. Sloane, Nov 18 2009

nonn

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[449*n^6/80 +1803*n^5/80 +713*n^4/16 +745*n^3/16 +1053*n^2/40 +37*n/5 +1: n in [0..30]]; // Vincenzo Librandi, Sep 17 2011

seq(coeff(series((1+147*x+1230*x^2+1885*x^3+714*x^4+63*x^5+x^6)/(1-x)^7, x,n+1),x,n),n=0..25); # Muniru A Asiru, Apr 29 2018

LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1, 154, 2287, 14695, 60907, 192493, 505912}, 30] (* G. C. Greubel, Apr 28 2018 *)
CoefficientList[Series[(1+147x+1230x^2+1885x^3+714x^4+63x^5+x^6)/(1-x)^7,{x,0,30}],x] (* Harvey P. Dale, Dec 30 2022 *)

x='x+O('x^30); Vec((1+147*x+1230*x^2+1885*x^3+714*x^4 +63*x^5 +x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

a(n) = 449*n^6/80 +1803*n^5/80 +713*n^4/16 +745*n^3/16 +1053*n^2/40 +37*n/5 +1. - R. J. Mathar, Sep 11 2011

A160841 Expansion of (1+147x+1230x^2+1915x^3+744x^4+66*x^5+x^6)/(1-x)^7.

Original entry on oeis.org

1, 154, 2287, 14725, 61147, 193546, 509293, 1172305, 2438317, 4684258, 8441731, 14434597, 23620663, 37237474, 56852209, 84415681, 122320441, 173462986, 241310071, 329969125, 444262771, 589807450, 773096149, 1001585233

Offset: 0

N. J. A. Sloane, Nov 18 2009

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[1+3*n*(n+1)*(38*n^4+112*n^3+183*n^2+127*n+50)/20: n in [0..30]]; // Vincenzo Librandi, Sep 19 2011

seq(coeff(series((1+147*x+1230*x^2+1915*x^3+744*x^4+66*x^5+x^6)/(1-x)^7, x,n+1),x,n),n=0..25); # Muniru A Asiru, Apr 29 2018

CoefficientList[Series[(1+147x+1230x^2+1915x^3+744x^4+66x^5+x^6)/(1-x)^7, {x,0,30}], x] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1, 154, 2287,14725,61147,193546,509293},30] (* Harvey P. Dale, Feb 11 2015 *)

x='x+O('x^30); Vec((1+147*x+1230*x^2+1915*x^3+744*x^4+66*x^5 + x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

a(n) = 1+3*n*(n+1)*(38*n^4+112*n^3+183*n^2+127*n+50)/20. - R. J. Mathar, Sep 17 2011

a(0)=1, a(1)=154, a(2)=2287, a(3)=14725, a(4)=61147, a(5)=193546, a(6)=509293, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)- 7*a(n-6)+ a(n-7). - Harvey P. Dale, Feb 11 2015

A160853 Expansion of (1+147x+1230x^2+1925x^3+754x^4+67*x^5+x^6)/(1-x)^7.

Original entry on oeis.org

1, 154, 2287, 14735, 61227, 193897, 510420, 1175273, 2445121, 4698328, 8468593, 14482711, 23702459, 37370607, 57061054, 84733089, 122789777, 174140470, 242267443, 331296655, 446072635, 592237493, 776314056, 1005793129

Offset: 0

N. J. A. Sloane, Nov 18 2009

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[1 +n*(n+1)*(1375*n^4+4022*n^3+6573*n^2+4582*n+1808)/240: n in [0..30]]; // Vincenzo Librandi, Sep 20 2011

seq(coeff(series((1+147*x+1230*x^2+1925*x^3+754*x^4+67*x^5+x^6)/(1-x)^7, x,n+1),x,n),n=0..25); # Muniru A Asiru, Apr 29 2018

LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1, 154, 2287, 14735, 61227, 193897, 510420}, 40] (* G. C. Greubel, Apr 28 2018 *)

x='x+O('x^30); Vec((1+147*x+1230*x^2+1925*x^3+754*x^4+67*x^5 + x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

a(n) = 1 +n*(n+1)*(1375*n^4+4022*n^3+6573*n^2+4582*n+1808)/240. - R. J. Mathar, Sep 17 2011

A160854 Expansion of (1+147x+1098x^2+1638x^3+632x^4+59*x^5+x^6)/(1-x)^7.

Original entry on oeis.org

1, 154, 2155, 13524, 55400, 173911, 455120, 1043547, 2164267, 4148584, 7463281, 12743446, 20828874, 32804045, 50041678, 74249861, 107522757, 152394886, 211898983, 289627432, 389797276, 517318803, 677867708, 877960831, 1125035471

Offset: 0

N. J. A. Sloane, Nov 18 2009

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[149*n^6/30 +81*n^5/4 +337*n^4/8 +142*n^3/3 +3529*n^2/120 +107 *n/12 +1: n in [0..30]]; // Vincenzo Librandi, Sep 20 2011

seq(coeff(series((1+147*x+1098*x^2+1638*x^3+632*x^4+59*x^5+x^6)/(1-x)^7, x,n+1),x,n),n=0..25); # Muniru A Asiru, Apr 29 2018

LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1, 154, 2155, 13524, 55400, 173911, 455120}, 30] (* G. C. Greubel, Apr 28 2018 *)

x='x+O('x^30); Vec((1+147*x+1098*x^2+1638*x^3+632*x^4+59*x^5 + x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

a(n) = 149*n^6/30 +81*n^5/4 +337*n^4/8 +142*n^3/3 +3529*n^2/120 +107*n/12 +1. - R. J. Mathar, Sep 17 2011

A160863 Expansion of (1+147x+1142x^2+1717x^3+656x^4+60*x^5+x^6)/(1-x)^7.

Original entry on oeis.org

1, 154, 2199, 13911, 57209, 179988, 471675, 1082509, 2246545, 4308382, 7753615, 13243011, 21650409, 34104344, 52033395, 77215257, 111829537, 158514274, 220426183, 301304623, 405539289, 538241628, 705319979, 913558437, 1170699441

Offset: 0

N. J. A. Sloane, Nov 18 2009

Source: the De Loera et al. article and the Haws website listed in A160747.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

[931*n^6/180 +1261*n^5/60 +1547*n^4/36 +565*n^3/12 +1276*n^2/45 +42*n/5+1: n in [0..30]]; // Vincenzo Librandi, Sep 20 2011

seq(coeff(series((1+147*x+1142*x^2+1717*x^3+656*x^4+60*x^5+x^6)/(1-x)^7, x,n+1),x,n),n=0..25); # Muniru A Asiru, Apr 29 2018

LinearRecurrence[{7,-21,35,-35,21,-7,1}, {1, 154, 2199, 13911, 57209, 179988, 471675}, 30] (* G. C. Greubel, Apr 28 2018 *)

x='x+O('x^30); Vec((1+147*x+1142*x^2+1717*x^3+656*x^4+60*x^5+ x^6)/(1-x)^7) \\ G. C. Greubel, Apr 28 2018

a(n) = 931*n^6/180 +1261*n^5/60 +1547*n^4/36 +565*n^3/12 +1276*n^2/45 +42*n/5 +1. - R. J. Mathar, Sep 17 2011

A355010 Array read by ascending antidiagonals: T(n, k) is the number of n-core partitions with k corners.

Original entry on oeis.org

1, 3, 1, 6, 5, 1, 10, 16, 7, 1, 15, 40, 31, 9, 1, 21, 85, 105, 51, 11, 1, 28, 161, 295, 219, 76, 13, 1, 36, 280, 721, 771, 396, 106, 15, 1, 45, 456, 1582, 2331, 1681, 650, 141, 17, 1, 55, 705, 3186, 6244, 6083, 3235, 995, 181, 19, 1, 66, 1045, 5985, 15156, 19348, 13663, 5685, 1445, 226, 21, 1

Offset: 2

Stefano Spezia, Jun 15 2022

T(n, k) is also equal to the number of cornerless Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.3 and Proposition 3.4 at pp. 13 - 14 in Cho et al.).

In proposition 3.4 in Cho et al., the Narayana number is defined as N(k, i) = binomial(k, i)*binomial(k, i-1)/k, unlike A001263.

			The array begins:
    1,  1,   1,   1,    1,    1,    1,    1, ...
    3,  5,   7,   9,   11,   13,   15,   17, ...
    6, 16,  31,  51,   76,  106,  141,  181, ...
   10, 40, 105, 219,  396,  650,  995, 1445, ...
   15, 85, 295, 771, 1681, 3235, 5685, 9325, ...
   ...

Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022.

Cf. A000012 (n = 2), A001263, A005408 (n = 3), A005891 (n = 4), A006007, A063490 (n = 5), A160747 (n = 6), A161680 (k = 1), A355011.

T[n_,k_]:=Sum[Binomial[k,i]Binomial[k,i-1]Binomial[n+2(k-i),2k]/k,{i,Min[k,Floor[n/2]]}]; Flatten[Table[T[n-k+1,k],{n,2,12},{k,1,n-1}]]

T(n, k) = Sum_{i=1..min(k,floor(n/2))} N(k, i)*binomial(n+2*(k-i), 2*k), where N(k, i) = binomial(k, i)*binomial(k, i-1)/k. (See proposition 3.4 in Cho et al.).

T(n, 2) = A006007(n-1).

cp's OEIS Frontend

A160837 G.f.: (1+38*x+262*x^2+475*x^3+254*x^4+37*x^5+x^6)/(1-x)^7.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

A160838 G.f.: (1+38*x+263*x^2+484*x^3+263*x^4+38*x^5+x^6)/(1-x)^7.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

PARI

Formula

A160839 Expansion of (78+1116*x+3492*x^2+3237*x^3+927*x^4+72*x^5+x^6)/(1-x)^7.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

A160840 Expansion of (1+147*x+1230*x^2+1885*x^3+714*x^4+63*x^5+x^6)/(1-x)^7.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

A160841 Expansion of (1+147*x+1230*x^2+1915*x^3+744*x^4+66*x^5+x^6)/(1-x)^7.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

A160853 Expansion of (1+147*x+1230*x^2+1925*x^3+754*x^4+67*x^5+x^6)/(1-x)^7.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

Mathematica

PARI

Formula

A160854 Expansion of (1+147*x+1098*x^2+1638*x^3+632*x^4+59*x^5+x^6)/(1-x)^7.

Views

Author

Keywords

Comments

Links

Programs

Magma

Maple

A160837 G.f.: (1+38x+262x^2+475x^3+254x^4+37*x^5+x^6)/(1-x)^7.

A160838 G.f.: (1+38x+263x^2+484x^3+263x^4+38*x^5+x^6)/(1-x)^7.

A160839 Expansion of (78+1116x+3492x^2+3237x^3+927x^4+72*x^5+x^6)/(1-x)^7.

A160840 Expansion of (1+147x+1230x^2+1885x^3+714x^4+63*x^5+x^6)/(1-x)^7.

A160841 Expansion of (1+147x+1230x^2+1915x^3+744x^4+66*x^5+x^6)/(1-x)^7.

A160853 Expansion of (1+147x+1230x^2+1925x^3+754x^4+67*x^5+x^6)/(1-x)^7.

A160854 Expansion of (1+147x+1098x^2+1638x^3+632x^4+59*x^5+x^6)/(1-x)^7.

A160863 Expansion of (1+147x+1142x^2+1717x^3+656x^4+60*x^5+x^6)/(1-x)^7.