A262809 -id:A262809 - cp's OEIS frontend

A262810 Number of lattice paths from {n}^n to {0}^n using steps that decrement one or more components by one.

1, 1, 13, 16081, 5552351121, 1050740615666453461, 179349571255187154941191217629, 41020870889694863957061607086939138327565057, 17469051230066445323872793284679234619523576313653708533767425

Offset: 0

Alois P. Heinz, Oct 02 2015

			a(2) = 13: [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,1),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)].

Alois P. Heinz, Table of n, a(n) for n = 0..25

Main diagonal of A262809.

Cf. A316677.

Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^n, {i, 0, j}], {j, 0, n^2}], {n, 1, 10}]}] (* Vaclav Kotesovec, Mar 23 2016 *)

a(n)=sum(j=0,n^2,sum(i=0,j, (-1)^i*binomial(j, i)*binomial(j - i, n)^n)) \\ Charles R Greathouse IV, Jul 29 2016

a(n) = A262809(n,n).

a(n) ~ n^(n^2 - n/2 + 1) / (exp(1/12) * 2^(n + log(2)/24) * Pi^((n-1)/2) * log(2)^(n^2+1)). - Vaclav Kotesovec, Mar 23 2016

A263065 Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one or more components by one.

Original entry on oeis.org

1, 541, 2244361, 14638956721, 117029959485121, 1050740615666453461, 10169807398958450670001, 103746115308050354021387521, 1100327453912286201909924526081, 12024609569670508078686022988554381, 134565509066155510620216211257550349401

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Also, the number of alignments for 5 sequences of length n each (Slowinski 1998).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (terms 0..100 from Alois P. Heinz)
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522

Column k=5 of A262809.

With[{k = 5}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 15}]] (* Vaclav Kotesovec, Mar 22 2016 *)

Recurrence: (n-1)^2*n^4*(1484375*n^10 - 35625000*n^9 + 380459375*n^8 - 2379691875*n^7 + 9648658375*n^6 - 26483049785*n^5 + 49802335614*n^4 - 63319196190*n^3 + 52054873977*n^2 - 24970596338*n + 5304283784)*a(n) = (n-1)^2*(20417578125*n^14 - 530857031250*n^13 + 6241768390625*n^12 - 43891361846875*n^11 + 205684605242500*n^10 - 677508063221175*n^9 + 1612085561506345*n^8 - 2803601202034769*n^7 + 3564158318615391*n^6 - 3277539874902099*n^5 + 2131595379572790*n^4 - 942888582994608*n^3 + 265378603877984*n^2 - 41959963867392*n + 2757380659200)*a(n-1) + (5551562500*n^16 - 166546875000*n^15 + 2299855640625*n^14 - 19385476578125*n^13 + 111504090473125*n^12 - 463446487931900*n^11 + 1437445134614810*n^10 - 3387090699899014*n^9 + 6112545662650711*n^8 - 8450360220919608*n^7 + 8884444155685163*n^6 - 6993443486776441*n^5 + 4013714078940498*n^4 - 1609501825795072*n^3 + 420362394759120*n^2 - 62905338995616*n + 3977994685824)*a(n-2) + (1870312500*n^16 - 59850000000*n^15 + 882904671875*n^14 - 7959826712500*n^13 + 49013510712500*n^12 - 218196359173225*n^11 + 724960516804615*n^10 - 1829325996659659*n^9 + 3532586966500778*n^8 - 5219142662751755*n^7 + 5853433612256896*n^6 - 4902859151966701*n^5 + 2984426972027036*n^4 - 1263868309818152*n^3 + 346653353359072*n^2 - 54082532707344*n + 3532661100864)*a(n-3) - (n-3)^2*(7421875*n^14 - 207812500*n^13 + 2631125000*n^12 - 19925368750*n^11 + 100603166250*n^10 - 357324371050*n^9 + 917934587470*n^8 - 1726295128861*n^7 + 2377475157009*n^6 - 2372287254911*n^5 + 1675297653876*n^4 - 803613605640*n^3 + 244104664208*n^2 - 41262015600*n + 2856959424)*a(n-4) + (n-4)^4*(n-3)^2*(1484375*n^10 - 20781250*n^9 + 126631250*n^8 - 440391875*n^7 + 962896500*n^6 - 1373591410*n^5 + 1282871689*n^4 - 765049709*n^3 + 274306866*n^2 - 52342548*n + 3936312)*a(n-5). - Vaclav Kotesovec, Mar 22 2016

a(n) ~ c * d^n / (Pi^2 * n^2), where d = 13755.27190241150817120839544215413203... is the real root of the equation -1 + 5*d - 1260*d^2 - 3740*d^3 - 13755*d^4 + d^5 = 0 and c = 0.5698188923151523225906967169329722766951557573868... is the root of the equation -1 - 1600*c^2 - 896000*c^4 - 204800000*c^6 - 16384000000*c^8 + 52428800000*c^10 = 0. - Vaclav Kotesovec, Mar 22 2016

A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 3^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 21, 9, 1, 1, 219, 657, 27, 1, 1, 3045, 119241, 22869, 81, 1, 1, 52923, 40365873, 80850987, 836001, 243, 1, 1, 1103781, 21955523049, 747786838869, 60579666801, 31436181, 729, 1, 1, 26857659, 17512689629457, 14298291269335467, 16117269494868801, 48066954848379, 1204022961, 2187, 1

Offset: 0

Seiichi Manyama, May 27 2025

			Square array begins:
  1,  1,      1,           1,                 1, ...
  1,  3,     21,         219,              3045, ...
  1,  9,    657,      119241,          40365873, ...
  1, 27,  22869,    80850987,      747786838869, ...
  1, 81, 836001, 60579666801, 16117269494868801, ...

Columns k=0..2 give A000012, A000244, 3^n * A084768(n).
Rows n=0..1 give A000012, A032033.
Cf. A262809, A384362.

a(n, k) = sum(i=0, k*n, 3^i*sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));

A(n,k) = (1/4) * Sum_{j>=0} (3/4)^j * binomial(j,n)^k.

A263061 Number of lattice paths from {5}^n to {0}^n using steps that decrement one or more components by one.

Original entry on oeis.org

1, 1, 1683, 32193253, 3147728203035, 1050740615666453461, 939073157252309315848923, 1909946024633189859690880523893, 7868854300758955660834916406038038395, 60169662022264019813634467045726478557798101, 797656368265147949572521540584234236944835806750363

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Row n=5 of A262809.

With[{r = 5}, Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 12}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)

From Vaclav Kotesovec, Mar 23 2016: (Start)
a(n) ~ 5^(4*n+1/2) * n!^5 / (Pi^2 * n^2 * 2^(3*n+5) * 3^n * (log(2))^(5*n+1)).
a(n) ~ sqrt(Pi) * 5^(4*n+1/2) * n^(5*n+1/2) / (2^(3*n+5/2) * 3^n * exp(5*n) * (log(2))^(5*n+1)).
(End)

A263062 Number of lattice paths from {6}^n to {0}^n using steps that decrement one or more components by one.

Original entry on oeis.org

1, 1, 8989, 1538743249, 1887593866439485, 10169807398958450670001, 179349571255187154941191217629, 8508048612432263410111274212273801489, 943457762940832669626002608045124343895474045, 220079308019032269943223432841210561656944209845808241

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Row n=6 of A262809.

With[{r = 6}, Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 12}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)

a(n) ~ sqrt(6*Pi) * (6^5/5!)^n * n^(6*n+1/2) / (8 * exp(6*n) * (log(2))^(6*n+1)). - Vaclav Kotesovec, Mar 23 2016

A263063 Number of lattice paths from {8}^n to {0}^n using steps that decrement one or more components by one.

Original entry on oeis.org

1, 1, 265729, 3776339263873, 756051015055329306625, 1100327453912286201909924526081, 7835213566547395052871069325808866414849, 209691630817770382144439647416526247292909726379393, 17469051230066445323872793284679234619523576313653708533767425

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Row n=8 of A262809.

With[{r = 8}, Flatten[{1, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, r]^k, {i, 0, j}], {j, 0, k*r}], {k, 1, 10}]}]] (* Vaclav Kotesovec, Mar 22 2016 *)

a(n) ~ sqrt(8*Pi) * (8^7/7!)^n * n^(8*n+1/2) / (16 * exp(8*n) * (log(2))^(8*n+1)). - Vaclav Kotesovec, Mar 23 2016

A263066 Number of lattice paths from {n}^6 to {0}^6 using steps that decrement one or more components by one.

Original entry on oeis.org

1, 4683, 308682013, 35941784497263, 5402040231378569121, 939073157252309315848923, 179349571255187154941191217629, 36585008462723983824862891403150079, 7835213566547395052871069325808866414849, 1742079663955078309800553960117733249663480043

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Vaclav Kotesovec, Recurrence (of order 6)

Column k=6 of A262809.

With[{k = 6}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 15}]] (* Vaclav Kotesovec, Mar 22 2016 *)

a(n) ~ sqrt(c) * d^n / (Pi*n)^(5/2), where d = 296476.91626442008149098622814984912648229139426918084511... is the root of the equation 1 - 18*d - 5397*d^2 - 123696*d^3 + 321303*d^4 - 296478*d^5 + d^6 = 0 and c = 0.19491147281619801027873171908746401584984116403035035539868... is the root of the equation -1 - 4608*c - 7962624*c^2 - 6341787648*c^3 - 2283043553280*c^4 - 300578991243264*c^5 + 1603087953297408*c^6 = 0. - Vaclav Kotesovec, Mar 23 2016

A263067 Number of lattice paths from {n}^7 to {0}^7 using steps that decrement one or more components by one.

Original entry on oeis.org

1, 47293, 58514835289, 143743469278461361, 480086443888959812703121, 1909946024633189859690880523893, 8508048612432263410111274212273801489, 41020870889694863957061607086939138327565057, 209691630817770382144439647416526247292909726379393

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Vaclav Kotesovec, Recurrence (of order 7)

Column k=7 of A262809.

With[{k = 7}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 15}]] (* Vaclav Kotesovec, Mar 22 2016 *)

a(n) ~ sqrt(c) * d^n / (Pi*n)^3, where d = 7553550.61983382187210690975164995019966376572879... is the root of the equation -1 + 7*d - 24031*d^2 - 374521*d^3 - 24850385*d^4 + 17978709*d^5 - 7553553*d^6 + d^7 = 0 and c = 0.1137319057755565367034882185733003109119819... is the root of the equation -1 - 12544*c - 61816832*c^2 - 151057858560*c^3 - 189486977777664*c^4 - 113186888059191296*c^5 - 25353862925258850304*c^6 + 231806746745223774208*c^7 = 0. - Vaclav Kotesovec, Mar 23 2016

A263068 Number of lattice paths from {n}^8 to {0}^8 using steps that decrement one or more components by one.

Original entry on oeis.org

1, 545835, 14623910308237, 874531783382503604463, 74896283763383392805211587121, 7868854300758955660834916406038038395, 943457762940832669626002608045124343895474045, 124069835911824710311393852646151897334844371419287295

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Column k=8 of A262809.

With[{k = 8}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 10}]] (* Vaclav Kotesovec, Mar 22 2016 *)

a(n) ~ sqrt(c) * d^n / (Pi*n)^(7/2), where d = 222082591.60172024210290001176855308841678706675284935653958249024021852... is the root of the equation 1 - 24*d - 102692*d^2 - 9298344*d^3 + 536208070*d^4 - 7106080680*d^5 - 1688209700*d^6 - 222082584*d^7 + d^8 = 0 and c = 0.065002105820899877029614597832047121767362853... . - Vaclav Kotesovec, Mar 23 2016

A263069 Number of lattice paths from {n}^9 to {0}^9 using steps that decrement one or more components by one.

Original entry on oeis.org

1, 7087261, 4659168491711401, 7687300579969605991710001, 19133358944433370977791260580721121, 60169662022264019813634467045726478557798101, 220079308019032269943223432841210561656944209845808241, 894709632166224106718347951886305028154659386016685862593012481

Offset: 0

Alois P. Heinz, Oct 08 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Column k=9 of A262809.

With[{k = 9}, Table[Sum[Sum[(-1)^i*Binomial[j, i]*Binomial[j - i, n]^k, {i, 0, j}], {j, 0, k*n}], {n, 0, 10}]] (* Vaclav Kotesovec, Mar 22 2016 *)

a(n) ~ sqrt(c) * d^n / (Pi*n)^4, where d = 7400694480.0494436216324852038000444393262965328... and c = 0.0365684849906610318536810681059888603001404... . - Vaclav Kotesovec, Mar 23 2016

cp's OEIS Frontend

A262810 Number of lattice paths from {n}^n to {0}^n using steps that decrement one or more components by one.

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A263065 Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one or more components by one.

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A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..k*n} 3^i * Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.

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A263061 Number of lattice paths from {5}^n to {0}^n using steps that decrement one or more components by one.

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A263062 Number of lattice paths from {6}^n to {0}^n using steps that decrement one or more components by one.

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A263063 Number of lattice paths from {8}^n to {0}^n using steps that decrement one or more components by one.

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A263066 Number of lattice paths from {n}^6 to {0}^6 using steps that decrement one or more components by one.

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A263067 Number of lattice paths from {n}^7 to {0}^7 using steps that decrement one or more components by one.

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Mathematica

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A263068 Number of lattice paths from {n}^8 to {0}^8 using steps that decrement one or more components by one.

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A384364 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{i=0..kn} 3^i Sum_{j=0..i} (-1)^j * binomial(i,j) * binomial(i-j,n)^k.