A227578 -id:A227578 - cp's OEIS frontend

A227583 Number of lattice paths from {n}^4 to {0}^4 using steps that decrement one component such that for each point (p_1,p_2,...,p_4) we have p_1<=p_2<=...<=p_4.

1, 1, 42, 3532, 456033, 77767945, 16104165970, 3848596333400, 1026843977181745, 298985252352030713, 93462550593036735356, 30988255904733937513266, 10797084830552485796491313, 3924879717822914648655364113, 1479988943331198988162565625314

Offset: 0

Alois P. Heinz, Jul 16 2013

nonn

			a(0) = 1: [(0,0,0,0)].
a(1) = 1: [(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1),(0,0,0,0)].
a(2) = 42: [(2,2,2,2),(0,2,2,2),(0,0,2,2),(0,0,0,2),(0,0,0,0)], ..., [(2,2,2,2),(1,2,2,2),(1,1,2,2),(1,1,1,2),(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1),(0,0,0,0)].

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=4 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([n$4])):
seq(a(n), n=0..16);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 1, 42, 3532][n+1], (
      (n+2)*(176594184*n^7-1252171596*n^6+3635870170*n^5
      -5627048591*n^4 +4990722596*n^3 -2520013645*n^2+664396290*n
      -69572160) *(n+1)^2* a(n-1) -3*(n-2)*(n+1)*(350669592*n^8
      -2376848760*n^7+6486604406*n^6-9505222590*n^5
      +8466195947*n^4-4746390780*n^3+1536942539*n^2-253906050*n
      +14945856) *a(n-2) +(n-3)*(1575354456*n^7-8934392532*n^6
      +18759259714*n^5-19551471603*n^4 +12013819684*n^3
      -4838385873*n^2+1019903146*n-96686592)*(n-2)^2 *a(n-3)
      -5000*(n-4)*(139932*n^4-226356*n^3+87227*n^2-21684*n-1919)
      *(n-3)^2 *(n-2)^3 *a(n-4)) / (2*(n+3)*(139932*n^4-786084*n^3
      +1605887*n^2-1434934*n+473280)*(n+2)^2*(n+1)^3))
    end:
seq(a(n), n=0..20);

b[l_] := b[l] = If[l[[-1]] == 0, 1, Sum[Sum[b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i - 1]]], l[[i]] - 1}], {i, 1, Length[l]}]];
a[n_] := b[Array[n&, 4]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A227578 *)

a(n) ~ 3*5^(4*n+11)/(2^13*Pi^(3/2)*(3*n)^(15/2)). - Vaclav Kotesovec, Jul 19 2013

A227584 Number of lattice paths from {4}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

Original entry on oeis.org

1, 8, 185, 7680, 456033, 34426812, 3086989927, 315051017342, 35566911169298, 4353511908566248, 569413385415535738, 78713723425497511522, 11403561640157735499129, 1719932910431380877877228, 268627766543783314569921051, 43259068400832620021992394382

Offset: 0

Alois P. Heinz, Jul 16 2013

nonn

			a(1) = 8: [(4),(0)], [(4),(1),(0)], [(4),(2),(0)], [(4),(2),(1),(0)], [(4),(3),(0)], [(4),(3),(1),(0)], [(4),(3),(2),(0)], [(4),(3),(2),(1),(0)].

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..422 (terms n=51..98 from Vaclav Kotesovec)

Row n=4 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([4$n])):
seq(a(n), n=0..16);

b[l_] := b[l] = If[l[[-1]] == 0, 1, Sum[Sum[b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i - 1]]], l[[i]] - 1}], {i, 1, Length[l]}]];
a[n_] := If[n == 0, 1, b[Array[4&, n]]];
a /@ Range[0, 16] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

From Vaclav Kotesovec, Nov 18 2016: (Start)
Recurrence: 2*(n+2)^3*(n+3)*(3*n + 4)*(3*n + 5)*(21546*n^6 - 45513*n^5 - 3699*n^4 + 13101*n^3 + 6745*n^2 + 2032*n + 228)*a(n) = (n+2)*(100641366*n^11 + 49968333*n^10 - 359008281*n^9 - 154345878*n^8 + 328506480*n^7 - 50516019*n^6 - 255412581*n^5 + 41482508*n^4 + 116397064*n^3 + 50875696*n^2 + 13946672*n + 1423680)*a(n-1) - 2*(173359116*n^12 + 510099768*n^11 - 127771911*n^10 - 1779706188*n^9 - 1320942528*n^8 + 1504634418*n^7 + 2094287811*n^6 + 74489810*n^5 - 783633768*n^4 - 274711936*n^3 + 13818000*n^2 + 12588128*n + 862080)*a(n-2) - 16*(2*n - 3)*(2*n + 1)*(3*n - 2)*(3*n - 1)*(4*n + 1)*(4*n + 3)*(21546*n^6 + 83763*n^5 + 91926*n^4 - 25905*n^3 - 108086*n^2 - 58260*n - 5560)*a(n-3).
a(n) ~ 2^(8*n+51/2) / (3^11 * Pi^(3/2) * n^(15/2)).
(End)

A227596 Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one component such that for each point (p_1,p_2,...,p_5) we have p_1<=p_2<=...<=p_5.

Original entry on oeis.org

1, 1, 132, 49100, 34426812, 36470203156, 51630369256916, 90650832149396184, 187978502469162658572, 445073778727031182727610, 1174228543974568589770758656, 3389786811049267225428045061056, 10559434352810002520295112134863908

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..116

Column k=5 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([n$5])):
seq(a(n), n=0..13);

b[l_] := b[l] = If[l[[-1]] == 0, 1, Sum[Sum[b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i - 1]]], l[[i]] - 1}], {i, 1, Length[l]}]];
a[n_] := If[n == 0, 1, b[Array[n&, 5]]];
a /@ Range[0, 13] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ 2*sqrt(5) * 6^(5*n+21) / (5^7 * 7^12 * Pi^2 * n^12). - Vaclav Kotesovec, Nov 20 2016

A227597 Number of lattice paths from {n}^6 to {0}^6 using steps that decrement one component such that for each point (p_1,p_2,...,p_6) we have p_1<=p_2<=...<=p_6.

Original entry on oeis.org

1, 1, 429, 750325, 3086989927, 22228291051255, 237791136700913751, 3418868469576233694591, 61845760669881132413037769, 1344481798162876850603732892817, 33976468300798036566458244068649205, 973569246761047672746215294808240044853

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..69

Column k=6 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([n$6])):
seq(a(n), n=0..13);

Conjecture: a(n) ~ 5 * 7^(6*n+29) / (2^58 * 3^8 * Pi^(5/2) * n^(35/2)). - Vaclav Kotesovec, Nov 20 2016

A227598 Number of lattice paths from {n}^7 to {0}^7 using steps that decrement one component such that for each point (p_1,p_2,...,p_7) we have p_1<=p_2<=...<=p_7.

Original entry on oeis.org

1, 1, 1430, 12310294, 315051017342, 16513520723284922, 1441565191975184121126, 184570140930218389159747070, 31862864761563509123808857974124, 6993293261428532974934599912795818724, 1869718376047919275097272876105318640045150

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..48

Column k=7 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([n$7])):
seq(a(n), n=0..12);

Conjecture: a(n) ~ 25 * sqrt(7) * 8^(7*n + 44) / (7^17 * 3^43 * Pi^3 * n^24). - Vaclav Kotesovec, Nov 21 2016

A227599 Number of lattice paths from {n}^8 to {0}^8 using steps that decrement one component such that for each point (p_1,p_2,...,p_8) we have p_1<=p_2<=...<=p_8.

Original entry on oeis.org

1, 1, 4862, 213446666, 35566911169298, 14323116388173517180, 10844768238749437970393066, 13220723286785303728967102618052, 23408169635197679203800470649923362577, 55994660641252674524946692511672567020920313, 171650174624972457949599385901886660192203614365332

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..37

Column k=8 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([n$8])):
seq(a(n), n=0..10);

Conjecture: a(n) ~ 42 * sqrt(5) * 9^(8*n + 58) / (8^20 * 10^29 * n^(63/2) * Pi^(7/2)). - Vaclav Kotesovec, Nov 26 2016

A227600 Number of lattice paths from {n}^9 to {0}^9 using steps that decrement one component such that for each point (p_1,p_2,...,p_9) we have p_1<=p_2<=...<=p_9.

Original entry on oeis.org

1, 1, 16796, 3868253164, 4353511908566248, 14071120934043157192832, 97106818062816381529413045436, 1190606938488172095512348078940830464, 22939433009552344381207995985855864376139032, 637028433009539403532335279417025047587902906655768

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..30

Column k=9 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([n$9])):
seq(a(n), n=0..10);

A227602 Number of lattice paths from {5}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

Original entry on oeis.org

1, 16, 1257, 238636, 77767945, 36470203156, 22228291051255, 16513520723284922, 14323116388173517180, 14071120934043157192832, 15313737501505148093502344, 18156604289232210133044514152, 23151467541948649805794187113781, 31425801906523386705389663813716908

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..60

Row n=5 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([5$n])):
seq(a(n), n=0..14);

b[l_] := b[l] = If[l[[-1]] == 0, 1, Sum[Sum[b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i - 1]]], l[[i]] - 1}], {i, 1, Length[l]}]];
a[n_] := If[n == 0, 1, b[Array[5&, n]]];
a /@ Range[0, 14] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ 9 * 5^(5*n + 41/2) / (2^37 * Pi^2 * n^12). - Vaclav Kotesovec, Nov 21 2016

A227603 Number of lattice paths from {6}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

Original entry on oeis.org

1, 32, 8925, 8285506, 16104165970, 51630369256916, 237791136700913751, 1441565191975184121126, 10844768238749437970393066, 97106818062816381529413045436, 1003769793669980634048599763674485, 11703712713157396870910671640141678850

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..40

Row n=6 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([6$n])):
seq(a(n), n=0..12);

Conjecture: a(n) ~ 2^(5/2) * 6^(6*n + 67/2) / (5^29 * Pi^(5/2) * n^(35/2)). - Vaclav Kotesovec, Nov 21 2016

A227604 Number of lattice paths from {7}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

Original entry on oeis.org

1, 64, 65445, 312077474, 3848596333400, 90650832149396184, 3418868469576233694591, 184570140930218389159747070, 13220723286785303728967102618052, 1190606938488172095512348078940830464, 129559009610760457771091688202936893773393

Offset: 0

Alois P. Heinz, Jul 17 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..38

Row n=7 of A227578.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
a:= n-> `if`(n=0, 1, b([7$n])):
seq(a(n), n=0..11);

Conjecture: a(n) ~ 2^4 * 5^2 * 7^(7*n + 85/2) / (6^37 * Pi^3 * n^24). - Vaclav Kotesovec, Nov 23 2016

cp's OEIS Frontend

A227583 Number of lattice paths from {n}^4 to {0}^4 using steps that decrement one component such that for each point (p_1,p_2,...,p_4) we have p_1<=p_2<=...<=p_4.

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A227584 Number of lattice paths from {4}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

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A227596 Number of lattice paths from {n}^5 to {0}^5 using steps that decrement one component such that for each point (p_1,p_2,...,p_5) we have p_1<=p_2<=...<=p_5.

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A227597 Number of lattice paths from {n}^6 to {0}^6 using steps that decrement one component such that for each point (p_1,p_2,...,p_6) we have p_1<=p_2<=...<=p_6.

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A227598 Number of lattice paths from {n}^7 to {0}^7 using steps that decrement one component such that for each point (p_1,p_2,...,p_7) we have p_1<=p_2<=...<=p_7.

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A227599 Number of lattice paths from {n}^8 to {0}^8 using steps that decrement one component such that for each point (p_1,p_2,...,p_8) we have p_1<=p_2<=...<=p_8.

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A227600 Number of lattice paths from {n}^9 to {0}^9 using steps that decrement one component such that for each point (p_1,p_2,...,p_9) we have p_1<=p_2<=...<=p_9.

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A227602 Number of lattice paths from {5}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

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Mathematica

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A227603 Number of lattice paths from {6}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.

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