A369102
Expansion of (1/x) * Series_Reversion( x * ((1-x)^4-x^4) ).
Original entry on oeis.org
1, 4, 26, 204, 1772, 16408, 158752, 1585968, 16235472, 169423232, 1795611168, 19275231872, 209140483328, 2289981517312, 25271472702464, 280795784911616, 3138701648319744, 35270318924758016, 398215386792574464, 4515067063939210240, 51388662166213954560
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^4-x^4))/x)
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(5*n+3, n-4*k))/(n+1);
A369161
Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^4) ).
Original entry on oeis.org
1, 3, 15, 91, 613, 4410, 33190, 258129, 2058281, 16737259, 138268611, 1157197639, 9790774861, 83606543660, 719638883748, 6237175439640, 54386540912490, 476782443732437, 4199713449255749, 37151346765537606, 329914740292813170, 2939975733035070000
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^4))/x)
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(4*n-k+2, n-4*k))/(n+1);
A365268
G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + x^3*A(x)^2).
Original entry on oeis.org
1, 1, 2, 5, 15, 48, 160, 549, 1929, 6909, 25134, 92612, 344924, 1296376, 4910656, 18728645, 71857133, 277160183, 1074085446, 4180057725, 16329796959, 64014638564, 251734985808, 992788252700, 3925688845948, 15560762343388, 61818928594952
Offset: 0
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a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*n-4*k+1, n-3*k)/(2*n-4*k+1));
A367317
Expansion of (1/x) * Series_Reversion( x * (1-x-x^4/(1-x)) ).
Original entry on oeis.org
1, 1, 2, 5, 15, 50, 177, 649, 2436, 9307, 36080, 141610, 561732, 2248709, 9073415, 36863549, 150676275, 619169360, 2556446520, 10600160707, 44121921044, 184291848864, 772204252280, 3244999395406, 13672564904027, 57749354647408, 244469827514066
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x^4/(1-x)))/x)
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(2*n-2*k, n-4*k))/(n+1);
A367414
Expansion of (1/x) * Series_Reversion( x * (1-x-x^4/(1-x)^2) ).
Original entry on oeis.org
1, 1, 2, 5, 15, 51, 187, 715, 2800, 11138, 44846, 182476, 749566, 3105575, 12966165, 54505650, 230508612, 980045835, 4186600220, 17960356014, 77343359518, 334217730014, 1448771849516, 6298222363395, 27452466169243, 119949953637406, 525284132440963
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x^4/(1-x)^2))/x)
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(2*n-k, n-4*k))/(n+1);
A368932
Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^4) ).
Original entry on oeis.org
1, 2, 7, 30, 144, 741, 3996, 22287, 127495, 743941, 4410555, 26492349, 160875186, 986007700, 6091548256, 37894543413, 237168491610, 1492323419929, 9434943086870, 59906035386393, 381832957589226, 2442251022673595, 15670578495195870
Offset: 0
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(3*n-3*k+1, n-4*k))/(n+1);
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^4))/x)
A369160
Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^4) ).
Original entry on oeis.org
1, 2, 7, 30, 144, 742, 4012, 22458, 129035, 756602, 4509141, 27233726, 166320987, 1025356360, 6372494608, 39882831334, 251146002084, 1590079213920, 10115878798130, 64634124182670, 414578955678690, 2668578654593970, 17232252926468640, 111602332042716450
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^4))/x)
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(3*n-2*k+1, n-4*k))/(n+1);
A367415
Expansion of (1/x) * Series_Reversion( x * (1-x-x^4/(1-x)^3) ).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 198, 793, 3255, 13529, 56696, 239340, 1017900, 4361840, 18828606, 81833505, 357865215, 1573549667, 6952392450, 30848928525, 137403484655, 614104910096, 2753200345000, 12378494389660, 55799811151140, 252141767612812, 1141894552992368
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x-x^4/(1-x)^3))/x)
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(2*n, n-4*k))/(n+1);
A383480
Number of lattice paths from (0,0) to (n,n) using steps (1,0),(4,0),(0,1).
Original entry on oeis.org
1, 2, 6, 20, 75, 294, 1176, 4752, 19350, 79310, 326898, 1353768, 5628441, 23478700, 98217840, 411879264, 1730924700, 7287941340, 30736775190, 129825892000, 549096132585, 2325216522420, 9857299586700, 41830206233400, 177673556967075, 755307883986084, 3213402383779812
Offset: 0
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f:= proc(x,y) option remember;
local t;
t:= 0;
if x >= 1 then t:= t + procname(x-1,y) fi;
if x >= 4 then t:= t + procname(x-4,y) fi;
if y >= 1 then t:= t + procname(x,y-1) fi;
t
end proc:
f(0,0):= 1:
seq(f(n,n),n=0..26); # Robert Israel, May 28 2025
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a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(2*n-3*k, n-4*k));
A383481
Coefficient of x^n in the expansion of 1 / (1-x-x^4)^n.
Original entry on oeis.org
1, 1, 3, 10, 39, 156, 630, 2556, 10431, 42823, 176748, 732810, 3049722, 12732188, 53299284, 223645200, 940355391, 3961092906, 16712516565, 70615352330, 298761296064, 1265504676810, 5366250376710, 22777466596560, 96768003904650, 411451657313931, 1750809473690436, 7455339422353396
Offset: 0
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f:= proc(n) local k; add(binomial(n+k-1,k)*binomial(2*n-3*k-1,n-4*k),k=0..n/4) end proc:
map(f, [$0..40]); # Robert Israel, May 28 2025
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a(n, s=4, t=1, u=0) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));
Showing 1-10 of 12 results.