Bryan T. Ek - cp's OEIS frontend

A302646 Values of unimodal polynomial analogous to A302612 and A302644 arising from a partition size <= 5 restriction.

0, 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 183755, 683046, 2443168, 8263360, 26184420, 77558760, 215182923, 561542454, 1385168400, 3245959640, 7260395142, 15567955260, 32124894880, 64016082000, 123566718600, 231661933776, 422854091037, 753068219386

Offset: 0

Bryan T. Ek, Apr 11 2018

Consider the unimodal polynomial from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=5. See G_5(n,k) from arXiv:1711.11252. If we make the simplification k=n and take the limit as q->1^-, we obtain the listed polynomial.

As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984.

			For n=6, G_5(6,6)=q^36+q^35+2*q^34+3*q^33+5*q^32+7*q^31+11*q^30+13*q^29+18*q^28+22*q^27+28*q^26+32*q^25+39*q^24+42*q^23+48*q^22+51*q^21+55*q^20+55*q^19+58*q^18+55*q^17+55*q^16+51*q^15+48*q^14+42*q^13+39*q^12+32*q^11+28*q^10+22*q^9+18*q^8+13*q^7+11*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the referenced paper). Then substituting q=1 yields 924.

Colin Barker, Table of n, a(n) for n = 0..1000
Bryan Ek, q-Binomials and related symmetric unimodal polynomials, arXiv:1711.11252 [math.CO], 2017-2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Cf. A000984, A002522, A302612, A302644, A302645.

concat(0, Vec(x*(1 - 13*x + 81*x^2 - 315*x^3 + 855*x^4 - 1701*x^5 + 2583*x^6 - 2961*x^7 + 2835*x^8 - 1365*x^9 + 2002*x^10) / (1 - x)^15 + O(x^40))) \\ Colin Barker, Apr 19 2018

a(n) = n*(n+3)*(n+2)*(n+1)*(n^10-50*n^9+1140*n^8-15420*n^7+136533*n^6-824370*n^5+3436190*n^4-9762880*n^3+18198936*n^2-20242080*n+10886400)/43545600.
From Colin Barker, Apr 19 2018: (Start)
G.f.: x*(1 - 13*x + 81*x^2 - 315*x^3 + 855*x^4 - 1701*x^5 + 2583*x^6 - 2961*x^7 + 2835*x^8 - 1365*x^9 + 2002*x^10) / (1 - x)^15.
a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15) for n>14.
(End)

More terms from Colin Barker, Apr 11 2018

0 prepended to the sequence and formulas adjusted accordingly by Colin Barker, Apr 19 2018

A302645 Values of unimodal polynomial analogous to A302612 and A302644 arising from a partition size <= 4 restriction.

Original entry on oeis.org

0, 1, 2, 6, 20, 70, 252, 924, 3432, 12705, 45430, 152438, 472836, 1352078, 3578680, 8827080, 20439984, 44745513, 93185994, 185640070, 355452020, 656846190, 1175604980, 2044130980, 3462303000, 5725877625, 9264588606, 14692562262, 22874204836, 35009334470

Offset: 0

Bryan T. Ek, Apr 10 2018

Consider the unimodal polynomial from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=4. See G_4(n,k) from arXiv:1711.11252. If we make the simplification k=n and take the limit as q->1^-, we obtain the listed polynomial.

As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984.

			For n=6, G_4(6,6)=q^36+q^35+2*q^34+3*q^33+5*q^32+7*q^31+11*q^30+13*q^29+18*q^28+22*q^27+28*q^26+32*q^25+39*q^24+42*q^23+48*q^22+51*q^21+55*q^20+55*q^19+58*q^18+55*q^17+55*q^16+51*q^15+48*q^14+42*q^13+39*q^12+32*q^11+28*q^10+22*q^9+18*q^8+13*q^7+11*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the referenced paper). Then substituting q=1 yields 924.

Colin Barker, Table of n, a(n) for n = 0..1000
Bryan Ek, q-Binomials and related symmetric unimodal polynomials, arXiv:1711.11252 [math.CO], 2017-2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000984, A002522, A302612, A302644, A302646.

concat(0, Vec(x*(1 - 10*x + 48*x^2 - 140*x^3 + 281*x^4 - 390*x^5 + 430*x^6 - 220*x^7 + 330*x^8) / (1 - x)^12 + O(x^40))) \\ Colin Barker, Apr 19 2018

a(n) = n*(n+2)*(n+1)*(n^8-32*n^7+462*n^6-3836*n^5+20013*n^4-66836*n^3+140804*n^2-171216*n+100800)/120960.
From Colin Barker, Apr 19 2018: (Start)
G.f.: x*(1 - 10*x + 48*x^2 - 140*x^3 + 281*x^4 - 390*x^5 + 430*x^6 - 220*x^7 + 330*x^8) / (1 - x)^12.
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12) for n>11.
(End)

More terms from Colin Barker, Apr 11 2018

0 prepended to the sequence and formulas adjusted accordingly by Colin Barker, Apr 19 2018

A302644 a(n) = (n+2)(n+1)(n^6-12n^5+70n^4-210n^3+409n^2-378*n+360)/720.

Original entry on oeis.org

1, 2, 6, 20, 70, 252, 896, 2976, 8955, 24310, 60038, 136500, 289016, 575680, 1087920, 1964384, 3408789, 5712426, 9282070, 14674100, 22635690, 34153988, 50514256, 73368000, 104812175, 147480606, 204648822, 280353556, 379528220, 508155720, 673440032, 883998016

Offset: 0

Bryan T. Ek, Apr 10 2018

The limit as q->1^- of the unimodal polynomial [q^(4k+3)(1-q^n)( q-q^n)-q^(3k)q(1+q)( 1-q^n)( q-q^2+q^5-q^n)-q^(2k)(q^(nk)(q^(2n)(q^(9)- q^(8)-q^(7)+q^(6)+ q^(5)-q^(3)+q)-q^n(q^(10)-q^(8)+q^(6)+q^(5))+q^(10))-q^(2n)+q^n(q^(5)+q^(4)-q^(2)+1)-q^(9)+q^(7)-q^(5)-q^(4)+q^(3)+q^(2)-q)+q^k q^(nk)q^(3) ( 1+q ) ( 1-q^n ) ( q^5-q^n+q^(n+3)-q^(n+4))-q^(nk)q^6(1-q^n)( q-q^n)]/[(1-q)^2(1-q^2)^2(1-q^3)(1-q^(n-1))(1-q^n)q^(2k+1)] after making the simplification k=n. The unimodal polynomial is from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=3. See G_3(n,k) from arXiv:1711.11252.

As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984.

			For n=4, G_3(4,4)=q^16+q^15+2*q^14+3*q^13+5*q^12+5*q^11+7*q^10+7*q^9+8*q^8+7*q^7+7*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the comments). Then substituting q=1 yields 70.

Colin Barker, Table of n, a(n) for n = 0..1000
Bryan Ek, q-Binomials and related symmetric unimodal polynomials, arXiv:1711.11252 [math.CO], 2017-2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000984, A002522, A302612, A302645, A302646.

Vec((1 - 7*x + 24*x^2 - 46*x^3 + 64*x^4 - 36*x^5 + 56*x^6) / (1 - x)^9 + O(x^40)) \\ Colin Barker, Apr 11 2018

a(n) = (n+2)*(n+1)*(n^6-12*n^5+70*n^4-210*n^3+409*n^2-378*n+360)/720.
From Colin Barker, Apr 11 2018: (Start)
G.f.: (1 - 7*x + 24*x^2 - 46*x^3 + 64*x^4 - 36*x^5 + 56*x^6) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.
(End)

More terms from Colin Barker, Apr 11 2018

A302612 a(n) = (n+1)(n^4-4n^3+11n^2-8n+12)/12.

Original entry on oeis.org

1, 2, 6, 20, 65, 186, 462, 1016, 2025, 3730, 6446, 10572, 16601, 25130, 36870, 52656, 73457, 100386, 134710, 177860, 231441, 297242, 377246, 473640, 588825, 725426, 886302, 1074556, 1293545, 1546890, 1838486, 2172512, 2553441, 2986050, 3475430, 4026996

Offset: 0

Bryan T. Ek, Apr 10 2018

The limit as q->1^- of the unimodal polynomial [q^(n*k+n+4)-q^(n*k+n+3)+q^(n*k+n+1)-q^(n*k+4)-q^((n-1)*k+n+3)+q^((n-1)*k+3)+q^(k+n+1)-q^(k+1)-q^n+q^3-q+1]/[(1-q)^2(1-q^2)(1-q^n)] after making the simplification k=n. This unimodal polynomial is from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=2. See G_2(n,k) from arXiv:1711.11252.

As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984.

			For n=4, G_2(4,4)=q^16+q^15+2*q^14+3*q^13+5*q^12+5*q^11+6*q^10+6*q^9+7*q^8+6*q^7+6*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the comments). Then substituting q=1 yields 65.

Colin Barker, Table of n, a(n) for n = 0..1000
Bryan Ek, q-Binomials and related symmetric unimodal polynomials, arXiv:1711.11252 [math.CO], 2017-2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000984, A002522, A302644, A302645, A302646.

Vec((1 - 4*x + 9*x^2 - 6*x^3 + 10*x^4) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Apr 11 2018

From Colin Barker, Apr 11 2018: (Start)
G.f.: (1 - 4*x + 9*x^2 - 6*x^3 + 10*x^4) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
(End)

More terms from Colin Barker, Apr 11 2018

A301381 Number of tied close American football games: number of ways for the game to end at the score of n to n and never be separated by more than one score after each play.

Original entry on oeis.org

1, 0, 2, 2, 6, 24, 80, 208, 922, 2310, 8794, 26000, 86632, 274120, 893552, 2837882, 9254642, 29470852, 95567342, 306155908, 987994256, 3174707284, 10228816628, 32893256236, 105937526030, 340778467916, 1097194416030, 3530389210580, 11364292475448, 36571646955122, 117713073900332

Offset: 0

Bryan T. Ek, Mar 20 2018

Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.

The same as counting walks that return to the x-axis of x-length n from the origin bounded above by y=8, below by y=-8, and using the steps {[2,2],[3,3],[8,4],[7,5],[6,6],[7,7],[8,8],[2,-2],[3,-3],[8,-4],[7,-5],[6,-6],[7,-7],[8,-8]}.

			There is no way to score 1 point so a(1)=0.
The number of ways to be tied at 4-4 is 6: there must be 2 safeties scored by each team which could be ordered in 4 choose 2 ways.
a(5)=24 since there must be 1 safety and 1 field goal for each team and there are 4! ways to order them.
a(n<=8) is fairly easy to compute since the bounds do not come into effect.

Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A300998, A301379, A301380.

taylor((16*t^29-16*t^28-56*t^27+52*t^26+100*t^25-52*t^24-136*t^23+108*t^22+66*t^21-71*t^20+134*t^19-5*t^18-320*t^17+50*t^16+78*t^15-47*t^14+60*t^13+78*t^12-158*t^11-8*t^10+31*t^8+t^7+37*t^6-t^5-10*t^4+2*t^3+6*t^2+t-1)/(32*t^33-112*t^32+24*t^31+324*t^30-300*t^29-40*t^28+52*t^27-542*t^26+784*t^25+766*t^24-1610*t^23+166*t^22+792*t^21-563*t^20+420*t^19+681*t^18-1320*t^17+190*t^16+246*t^15-87*t^14+74*t^13+304*t^12-380*t^11+6*t^10-10*t^9+25*t^8-25*t^7+85*t^6-3*t^5-22*t^4+2*t^3+8*t^2+t-1),t=0,N);

G.f.: (16*t^29-16*t^28-56*t^27+52*t^26+100*t^25-52*t^24-136*t^23+108*t^22+66*t^21-71*t^20+134*t^19-5*t^18-320*t^17+50*t^16+78*t^15-47*t^14+60*t^13+78*t^12-158*t^11-8*t^10+31*t^8+t^7+37*t^6-t^5-10*t^4+2*t^3+6*t^2+t-1)/(32*t^33-112*t^32+24*t^31+324*t^30-300*t^29-40*t^28+52*t^27-542*t^26+784*t^25+766*t^24-1610*t^23+166*t^22+792*t^21-563*t^20+420*t^19+681*t^18-1320*t^17+190*t^16+246*t^15-87*t^14+74*t^13+304*t^12-380*t^11+6*t^10-10*t^9+25*t^8-25*t^7+85*t^6-3*t^5-22*t^4+2*t^3+8*t^2+t-1).

A301380 Number of tied close American football games: number of ways for the game to have n scoring plays, never be separated by more than one score after each play, and be tied at the end.

Original entry on oeis.org

1, 0, 14, 90, 1114, 10718, 113216, 1152540, 11906042, 122269186, 1258639394, 12943924960, 133168371652, 1369830663678, 14091618522696, 144958402357534, 1491181759508514, 15339664777115086, 157798158205312580, 1623258461571800764, 16698349602838663718, 171774768145224952472

Offset: 0

Bryan T. Ek, Mar 20 2018

Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.

The same as counting walks that return to the x-axis of x-length n from the origin bounded above by y=8, below by y=-8, and using the steps {[1,8],..,[1,2],[1,-2],..,[1,-8]}.

			There are no tied games with 1 scoring play. To have tied games after 2 scoring plays requires each team to score the same number of points (7 possibilities) in each play (2 orderings): hence 14 walks.

Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A300998, A301379, A301381.

taylor((1-4*t-45*t^2-43*t^3+98*t^4+108*t^5-24*t^6-30*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9),t=0,N);

G.f.: (1-4*t-45*t^2-43*t^3+98*t^4+108*t^5-24*t^6-30*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9).

A301379 Number of close American football games: number of ways for the game to have n scoring plays and never be separated by more than one score after each play.

Original entry on oeis.org

1, 14, 128, 1378, 13932, 144300, 1480376, 15245184, 156756896, 1612836306, 16589928984, 170664508406, 1755592926518, 18059752212038, 185779058543356, 1911097952732140, 19659326724616886, 202234169412143472, 2080368880383488938, 21400612097499844490, 220146623069820835050

Offset: 0

Bryan T. Ek, Mar 19 2018

Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.

The same as counting walks of x-length n from the origin bounded above by y=8, below by y=-8, and using the steps {[1,8],..,[1,2],[1,-2],..,[1,-8]}.

			For n=1, any step is valid. For n=2, any walk with steps of opposite direction is valid while [[1,3],[1,6]] is an example of an invalid walk.

Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A300998, A301380, A301381.

taylor((1+10*t+13*t^2-37*t^3-40*t^4+28*t^5+26*t^6-2*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9),t=0,N);

G.f.: (1+10*t+13*t^2-37*t^3-40*t^4+28*t^5+26*t^6-2*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9).

A300998 Number of close American football games: number of ways for the game to end after n points have been scored and never be separated by more than one score after each play.

Original entry on oeis.org

1, 0, 2, 2, 4, 8, 14, 28, 52, 78, 156, 272, 520, 832, 1616, 2734, 5224, 8756, 16798, 28192, 54118, 90644, 173876, 292816, 561574, 938748, 1802188, 3031400, 5812998, 9734470, 18684588, 31367492, 60172174, 100893834, 193598664, 324824728, 623209036, 1045201398, 2005438304, 3364638978

Offset: 0

Bryan T. Ek, Mar 20 2018

Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.

The same as counting walks of x-length n from the origin bounded above by y=8, below by y=-8, and using the steps {[2,2],[3,3],[8,4],[7,5],[6,6],[7,7],[8,8],[2,-2],[3,-3],[8,-4],[7,-5],[6,-6],[7,-7],[8,-8]}.

			There is no way to score 1 point so a(1)=0.
There are 2 ways to score 2 or 3 points.
a(n<=8) is fairly easy to compute since the bounds do not come into effect.
a(9)=78. The unallowable walks are those with 9 points all of the same magnitude: [2,2,2,3],[3,3,3],[2,7],[3,6] (and the negatives and reorderings). A total of 18 unallowable walks. The total walks of length 9 are 2*4*2 (2 and 7 points and ordering) + 2*2*2 (3 and 6) + 2*2*2 (3 and 3 and 3) + 2*2*2*2*4 (2 and 2 and 2 and 3). The total is then 16+8+8+64-18=78.

Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A301379, A301380, A301381.

taylor(-(16*t^58-16*t^57-48*t^56+56*t^55-20*t^54-8*t^53+168*t^52-164*t^51-32*t^50+104*t^49-128*t^48+96*t^47-64*t^46+52*t^45-188*t^44+66*t^43+350*t^42-352*t^41+421*t^40-160*t^39-606*t^38+540*t^37-145*t^36-54*t^35+234*t^34-26*t^33-56*t^32-162*t^31+334*t^30-200*t^29+107*t^28-18*t^27-388*t^26+352*t^25-94*t^24-34*t^23+136*t^22-54*t^21+48*t^20-112*t^19+64*t^18-8*t^17+7*t^16+40*t^15-81*t^14+62*t^13-71*t^12-2*t^11+31*t^10-18*t^9+24*t^8+4*t^7-8*t^6+6*t^5-6*t^4+2*t^3+t^2+1)/(32*t^66-112*t^64+24*t^62+324*t^60-300*t^58-40*t^56+52*t^54-542*t^52+784*t^50+766*t^48-1610*t^46+166*t^44+792*t^42-563*t^40+420*t^38+681*t^36-1320*t^34+190*t^32+246*t^30-87*t^28+74*t^26+304*t^24-380*t^22+6*t^20-10*t^18+25*t^16-25*t^14+85*t^12-3*t^10-22*t^8+2*t^6+8*t^4+t^2-1),t=0,N);

G.f.: -(16*t^58-16*t^57-48*t^56+56*t^55-20*t^54-8*t^53+168*t^52-164*t^51-32*t^50+104*t^49-128*t^48+96*t^47-64*t^46+52*t^45-188*t^44+66*t^43+350*t^42-352*t^41+421*t^40-160*t^39-606*t^38+540*t^37-145*t^36-54*t^35+234*t^34-26*t^33-56*t^32-162*t^31+334*t^30-200*t^29+107*t^28-18*t^27-388*t^26+352*t^25-94*t^24-34*t^23+136*t^22-54*t^21+48*t^20-112*t^19+64*t^18-8*t^17+7*t^16+40*t^15-81*t^14+62*t^13-71*t^12-2*t^11+31*t^10-18*t^9+24*t^8+4*t^7-8*t^6+6*t^5-6*t^4+2*t^3+t^2+1)/(32*t^66-112*t^64+24*t^62+324*t^60-300*t^58-40*t^56+52*t^54-542*t^52+784*t^50+766*t^48-1610*t^46+166*t^44+792*t^42-563*t^40+420*t^38+681*t^36-1320*t^34+190*t^32+246*t^30-87*t^28+74*t^26+304*t^24-380*t^22+6*t^20-10*t^18+25*t^16-25*t^14+85*t^12-3*t^10-22*t^8+2*t^6+8*t^4+t^2-1).

A300390 The number of paths of length 7n from the origin to the line y = 3x/4 with unit east and north steps that stay below the line or touch it.

Original entry on oeis.org

1, 5, 227, 15090, 1182187, 101527596, 9247179818, 877362665128, 85783306955099, 8582893111512001, 874542924575207352, 90437361732467946334, 9467275300762187682554, 1001309098267187214993056, 106836493655355495755649544, 11485688815900189437990930096, 1242964338344397490958154292155

Offset: 0

Bryan T. Ek, Mar 05 2018

Equivalent to nonnegative walks from (0,0) to (7*n,0) with step set [1,3], [1,-4].

			For n=1, the possible walks are EEEENNN, EEENENN, EENEENN, EEENNEN, EENENEN.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A001764, A060941, A300386, A300387, A300388, A300389, A300391.

m = 17; f = 0; Do[f = f^35*t^5 - f^31*t^4 + f^30*t^4 - f^29*t^4 + 5*f^28*t^4 - f^25*t^3 + f^24*t^3 + 3*f^23*t^3 - 4*f^22*t^3 + 10*f^21*t^3 + f^19*t^2 - f^18*t^2 + 5*f^17*t^2 + 3*f^16*t^2 - 6*f^15*t^2 + 10*f^14*t^2 + f^13*t - f^12*t + 3*f^10*t + f^9*t - 4*f^8*t + 5*f^7*t + 1 + O[t]^m, {m}]; CoefficientList[f, t] (* Jean-François Alcover, Feb 18 2019 *)

G.f. satisfies: f = f^35*t^5 - f^31*t^4 + f^30*t^4 - f^29*t^4 + 5*f^28*t^4 - f^25*t^3 + f^24*t^3 + 3*f^23*t^3 - 4*f^22*t^3 + 10*f^21*t^3 + f^19*t^2 - f^18*t^2 + 5*f^17*t^2 + 3*f^16*t^2 - 6*f^15*t^2 + 10*f^14*t^2 + f^13*t - f^12*t + 3*f^10*t + f^9*t - 4*f^8*t + 5*f^7*t + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/7)*binomial(7*n, 3*n)*x^n/n ) - Bizley.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/7)*binomial(7*n-7*k, 3*n-3*k)*a(k) for n >= 1. (End)

A300391 The number of paths of length 8n from the origin to the line y = 3x/5 with unit east and north steps that stay below the line or touch it.

Original entry on oeis.org

1, 7, 525, 58040, 7574994, 1084532963, 164734116407, 26070940600055, 4252443527211637, 709846349042619913, 120679177855928146859, 20822762876863605793639, 3637213213067542990001936, 641912742432770594132245835, 114287840570892852593437353124, 20502971288127330644273350110698

Offset: 0

Bryan T. Ek, Mar 05 2018

Equivalent to nonnegative walks from (0,0) to (8*n,0) with step set [1,3], [1,-5].

			For n=1, the possible walks are EEEEENNN, EEEENENN, EEEENNEN, EEENEENN, EEENENEN, EENEEENN, EENEENEN.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A001764, A060941, A300386, A300387, A300388, A300389, A300390.

G.f. f satisfies f = t^7*f^56 - 2*t^6*f^51 + t^6*f^50 - t^6*f^49 + 7*t^6*f^48 + t^5*f^46 - t^5*f^45 - 3*t^5*f^43 + 5*t^5*f^42 - 6*t^5*f^41 + 21*t^5*f^40 - 3*t^4*f^37 - 3*t^4*f^36 + 8*t^4*f^35 + 10*t^4*f^34 - 15*t^4*f^33 + 35*t^4*f^32 - 2*t^3*f^31 + 2*t^3*f^30 - 9*t^3*f^28 + 22*t^3*f^27 + 10*t^3*f^26 - 20*t^3*f^25 + 35*t^3*f^24 + 3*t^2*f^22 + 5*t^2*f^21 - 9*t^2*f^20 + 18*t^2*f^19 + 5*t^2*f^18 - 15*t^2*f^17 + t*(21*t + 1)*f^16 - t*f^15 + 3*t*f^13 - 3*t*f^12 + 5*t*f^11 + t*f^10 - 6*t*f^9 + 7*t*f^8 + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/8)*binomial(8*n, 3*n)*x^n/n ) - Bizley.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/8)*binomial(8*n-8*k, 3*n-3*k)*a(k) for n >= 1. (End)

cp's OEIS Frontend

User: Bryan T. Ek

A302646 Values of unimodal polynomial analogous to A302612 and A302644 arising from a partition size <= 5 restriction.

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A302645 Values of unimodal polynomial analogous to A302612 and A302644 arising from a partition size <= 4 restriction.

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A302644 a(n) = (n+2)*(n+1)*(n^6-12*n^5+70*n^4-210*n^3+409*n^2-378*n+360)/720.

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A302612 a(n) = (n+1)*(n^4-4*n^3+11*n^2-8*n+12)/12.

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A301381 Number of tied close American football games: number of ways for the game to end at the score of n to n and never be separated by more than one score after each play.

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A301380 Number of tied close American football games: number of ways for the game to have n scoring plays, never be separated by more than one score after each play, and be tied at the end.

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A301379 Number of close American football games: number of ways for the game to have n scoring plays and never be separated by more than one score after each play.

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A302644 a(n) = (n+2)(n+1)(n^6-12n^5+70n^4-210n^3+409n^2-378*n+360)/720.

A302612 a(n) = (n+1)(n^4-4n^3+11n^2-8n+12)/12.

A300390 The number of paths of length 7n from the origin to the line y = 3x/4 with unit east and north steps that stay below the line or touch it.

A300391 The number of paths of length 8n from the origin to the line y = 3x/5 with unit east and north steps that stay below the line or touch it.