A177447
G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2) = 1+x.
Original entry on oeis.org
1, 1, 1, 3, 18, 172, 2313, 40626, 887326, 23282964, 715540140, 25259729071, 1008721104654, 45008479039824, 2221170817590696, 120209722115431950, 7083266027910364710, 451620678137942740132, 30990400538494184551692, 2277988537997377457967690, 178626191260072536476398000
Offset: 0
1+x = 1 + 1*x/(1+x) + 1*x^2/(1+x)^4 + 3*x^3/(1+x)^9 + 18*x^4/(1+x)^16 + 172*x^5/(1+x)^25 + 2313*x^6/(1+x)^36 +...
Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 2n+1 times, starting with a '1' in row 0, as illustrated by:
1;
1, 1, 1;
1, 2, 3, 3, 3, 3, 3;
1, 3, 6, 9, 12, 15, 18, 18, 18, 18, 18, 18, 18;
1, 4, 10, 19, 31, 46, 64, 82, 100, 118, 136, 154, 172, 172, 172, 172, 172, 172, 172, 172, 172;
1, 5, 15, 34, 65, 111, 175, 257, 357, 475, 611, 765, 937, 1109, 1281, 1453, 1625, 1797, 1969, 2141, 2313, 2313, 2313, 2313, 2313, 2313, 2313, 2313, 2313; ...
-
a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
*(-1)^(n-j)*binomial(1 +j*(j-1), n-j), j=0..n-1))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Jul 10 2022
-
{a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(k^2)),n)}
-
{A=[1,1];for(i=1,40,A=concat(A,-Vec(sum(n=0,#A-1,A[n+1]*x^n/(1+x+x*O(x^#A))^(n^2)))[#A+1]));for(n=0,#A-1,print1(A[n+1],", "))}
A133316
G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(n^2).
Original entry on oeis.org
1, 1, 2, 8, 54, 544, 7508, 133704, 2943194, 77589536, 2391477804, 84582890704, 3382005372970, 151034046369696, 7458091839548356, 403808650013237224, 23801728042233670770, 1517930142778063770304, 104179592763803229618620
Offset: 0
1/(1-x) = 1 + x/(1+x) + 2*x^2/(1+x)^4 + 8*x^3/(1+x)^9 + 54*x^4/(1+x)^16 +...
-
{a(n)=if(n==0,1,polcoeff(-(1-x)*sum(m=0,n-1,a(m)*x^m/(1+x +x*O(x^n))^(m^2)),n))}
-
{a(n)=if(n==0, 1, 1 - sum(j=0, n-1, a(j)*(-1)^(n-j)*binomial(j^2+n-j-1, n-j)))}
A215241
Unsigned matrix inverse of triangle A214398, as a triangle read by rows n >= 1.
Original entry on oeis.org
1, 1, 1, 3, 4, 1, 18, 26, 9, 1, 172, 256, 99, 16, 1, 2313, 3489, 1416, 264, 25, 1, 40626, 61696, 25650, 5120, 575, 36, 1, 887326, 1352518, 569772, 117980, 14450, 1098, 49, 1, 23282964, 35566368, 15099042, 3193728, 410850, 34608, 1911, 64, 1, 715540140, 1094499820, 466865280, 100049120, 13259705, 1186857, 73696, 3104, 81, 1
Offset: 1
Triangle begins:
1;
1, 1;
3, 4, 1;
18, 26, 9, 1;
172, 256, 99, 16, 1;
2313, 3489, 1416, 264, 25, 1;
40626, 61696, 25650, 5120, 575, 36, 1;
887326, 1352518, 569772, 117980, 14450, 1098, 49, 1;
23282964, 35566368, 15099042, 3193728, 410850, 34608, 1911, 64, 1;
...
The matrix inverse is a signed version of triangle A214398:
1;
-1, 1;
1, -4, 1;
-1, 10, -9, 1;
1, -20, 45, -16, 1;
-1, 35, -165, 136, -25, 1;
1, -56, 495, -816, 325, -36, 1;
-1, 84, -1287, 3876, -2925, 666, -49, 1; ...
in which the g.f. of column k is 1/(1+x)^(k^2) for k >= 1.
ILLUSTRATE G.F. OF COLUMNS:
k=1: 1 = 1/(1+x) + 1*x/(1+x)^4 + 3*x^2/(1+x)^9 + 18*x^3/(1+x)^16 + 172*x^4/(1+x)^25 + 2313*x^5/(1+x)^36 + 40626*x^6/(1+x)^49 + ...
k=2: 1 = 1/(1+x)^4 + 4*x/(1+x)^9 + 26*x^2/(1+x)^16 + 256*x^3/(1+x)^25 + 3489*x^4/(1+x)^36 + 61696*x^5/(1+x)^49 + ...
k=3: 1 = 1/(1+x)^9 + 9*x/(1+x)^16 + 99*x^2/(1+x)^25 + 1416*x^3/(1+x)^36 + 25650*x^4/(1+x)^49 + ...
k=4: 1 = 1/(1+x)^16 + 16*x/(1+x)^25 + 264*x^2/(1+x)^36 + 5120*x^3/(1+x)^49 + ...
Cf.
A214398 (unsigned matrix inverse).
-
T[n_, k_] := Module[{M}, M = Table[Binomial[c^2 + r - c - 1, r - c], {r, 1, n}, {c, 1, n}]; (-1)^(n - k) Inverse[M][[n, k]]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 05 2023, after PARI program *)
-
{T(n, k)=local(M=matrix(n,n,r,c,binomial(c^2+r-c-1, r-c)));(-1)^(n-k)*(M^-1)[n,k]}
for(n=1, 12, for(k=1, n, print1(T(n, k), ", ")); print(""))
A215243
G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^((n+3)^2).
Original entry on oeis.org
1, 9, 99, 1416, 25650, 569772, 15099042, 466865280, 16545757617, 662459717350, 29611195466373, 1463138718427008, 79255793863426950, 4673128560507694980, 298096897542679853190, 20462949699720864598464, 1504570012129788012314910, 118004419030927157257862025
Offset: 0
G.f.: 1 = 1/(1+x)^9 + 9*x/(1+x)^16 + 99*x^2/(1+x)^25 + 1416*x^3/(1+x)^36 + 25650*x^4/(1+x)^49 +...
-
{a(n)=if(n==0,1,polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+x+x*O(x^n))^((k+3)^2)), n))}
for(n=0,21,print1(a(n)", "))
Showing 1-4 of 4 results.
Comments