A177447 -id:A177447 - cp's OEIS frontend

A133316 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(n^2).

1, 1, 2, 8, 54, 544, 7508, 133704, 2943194, 77589536, 2391477804, 84582890704, 3382005372970, 151034046369696, 7458091839548356, 403808650013237224, 23801728042233670770, 1517930142778063770304, 104179592763803229618620

Offset: 0

Paul D. Hanna, Dec 24 2010

nonn

			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 +...

Cf. A141761, A177447, A215241, A215242, A215243.

{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)))}

a(n) = 1 - Sum_{j=0..n-1} a(j) * (-1)^(n-j) * C(j^2 + n-j-1, n-j) for n>0, with a(0)=1.

A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

Original entry on oeis.org

1, 1, 2, 9, 70, 805, 12480, 245847, 5909338, 168310515, 5556486450, 209003251240, 8835266400450, 415094928861530, 21473740362658640, 1213683089969940075, 74446121738526773490, 4927385997649620215895, 350145746700442604768346

Offset: 0

Paul D. Hanna, May 09 2010

nonn

			1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^6 + 9*x^3/(1+x)^12 + 70*x^4/(1+x)^20 + 805*x^5/(1+x)^30 +...
1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^4 + 9*x^3*(1-x)^9 + 70*x^4*(1-x)^16 + 805*x^5*(1-x)^25 +...
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 2(n+1) times, starting with a '1' in row 0, as illustrated by:
  1;
  1, 1;
  1, 2,  2,  2,  2;
  1, 3,  5,  7,  9,  9,   9,   9,   9,   9;
  1, 4,  9, 16, 25, 34,  43,  52,  61,  70,  70,  70,  70,  70,  70,  70,  70;
  1, 5, 14, 30, 55, 89, 132, 184, 245, 315, 385, 455, 525, 595, 665, 735, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..356

Cf. A107877, A177447, A177448, A177449, A209440.

a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial(1+ j^2, n-j), j=0..n-1))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 08 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*(k+1))),n)}

G.f.: Sum_{n>=0} a(n)*x^n*(1-x)^(n^2) = 1/(1-x).
G.f.: Sum_{n>=0} a(n)*x^n*C(-x)^(n^2+2n) = 1/C(-x) where C(x) is the Catalan function of A000108.
a(n) = number of subpartitions of partition consisting of the first n square numbers starting with zero for n>0; e.g., a(4) = subp([0,1,4,9]) = 70. See A115728 for the definition of subpartitions.

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

Paul D. Hanna, Aug 06 2012

			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 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..1081

Cf. A177447 (column 1), A215242 (column 2), A215243 (column 3); A133316 (row sums).

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(""))

G.f.: x*y/(1-x*y) = Sum_{n>=1} Sum_{k=1..n} T(n,k)*x^n*y^k/(1+x)^(n^2).
G.f. of column k: 1 = Sum_{n>=k} T(n,k)*x^(n-k)/(1+x)^(n^2).
Column 1 forms A177447.
Row sums form A133316.

A215242 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^((n+2)^2).

Original entry on oeis.org

1, 4, 26, 256, 3489, 61696, 1352518, 35566368, 1094499820, 38670814348, 1545160614694, 68970980789472, 3404652821768232, 184295822142051600, 10861040169788302030, 692560292664515634112, 47527552597795293035916, 3493783983256399634130360, 273974326317024551368217200

Offset: 0

Paul D. Hanna, Aug 06 2012

nonn

Column 2 of triangle A215241.

			G.f.: 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 +...
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 2*n+1 times, starting with '[1,1,1,1]' in row 1, as illustrated by:
1, 1,  1,  1;
1, 2,  3,  4,  4,   4,   4,   4;
1, 3,  6, 10, 14,  18,  22,  26,  26,  26,  26,  26,  26,   26;
1, 4, 10, 20, 34,  52,  74, 100, 126, 152, 178, 204, 230,  256,  256,  256,  256,  256,  256,  256,  256,  256;
1, 5, 15, 35, 69, 121, 195, 295, 421, 573, 751, 955, 1185, 1441, 1697, 1953, 2209, 2465, 2721, 2977, 3233, 3489, 3489, 3489, 3489, 3489, 3489, 3489, 3489, 3489; ...

Cf. A215241, A177447, A215243, A133316.

{a(n)=if(n==0,1,polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+x+x*O(x^n))^((k+2)^2)), n))}
for(n=0,21,print1(a(n)", "))

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

Paul D. Hanna, Aug 06 2012

nonn

Column 3 of triangle A215241.

			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 +...

Cf. A215241, A177447, A215242, A133316.

{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)", "))

A177448 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(2n^2) = 1+x.

Original entry on oeis.org

1, 1, 2, 13, 166, 3324, 92718, 3354712, 150206430, 8050991676, 504049958320, 36172232930282, 2931474921768206, 265078092222575572, 26480336590135734816, 2898139377307388441520, 345055687960080723910286

Offset: 0

Paul D. Hanna, May 09 2010

nonn

			1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^8 + 13*x^3/(1+x)^18 + 166*x^4/(1+x)^32 + 3324*x^5/(1+x)^50 + 92718*x^6/(1+x)^72 +...

Cf. A177447, A177449, A177450.

{a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(2*k^2)),n)}

a(n) = number of subpartitions of the partition [0,1,6,15,28,...,2(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.

A177449 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(3n^2) = 1+x.

Original entry on oeis.org

1, 1, 3, 30, 586, 17865, 756285, 41440056, 2805638310, 227131872654, 21459076173105, 2322336372705030, 283667666439112350, 38643426990067599005, 5813534115429573742587, 957883907138024944675200

Offset: 0

Paul D. Hanna, May 09 2010

nonn

			1+x = 1 + 1*x/(1+x)^3 + 3*x^2/(1+x)^12 + 30*x^3/(1+x)^27 + 586*x^4/(1+x)^48 + 17865*x^5/(1+x)^75 + 756285*x^6/(1+x)^108 +...

Cf. A177447, A177448, A177450.

{a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(3*k^2)),n)}

a(n) = number of subpartitions of the partition [0,2,10,24,44,...,3(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.

A229711 G.f.: Sum_{n>=0} a(n)*x^n / (1+x)^(n^3) = x.

Original entry on oeis.org

1, 1, 7, 154, 7329, 621054, 83287785, 16339143828, 4433073578739, 1595084475573057, 736780843688600494, 425703341782263982836, 301237142332910524156150, 256518292539312393631293756, 259004327874862610288497260501, 306183323229810278424153632807196

Offset: 1

Paul D. Hanna, Sep 27 2013

nonn

			G.f.: x = 1*x/(1+x) + 1*x^2/(1+x)^8 + 7*x^3/(1+x)^27 + 154*x^4/(1+x)^64 + 7329*x^5/(1+x)^125 + 621054*x^6/(1+x)^216 + 83287785*x^7/(1+x)^343 +...
ALTERNATE GENERATING METHOD.
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 3*n*(n-1)+1 times, starting with a '1' in row 1, as illustrated by:
1;
1, 1, 1, 1, 1, 1, 1;
1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7;
1, 3, 6, 10, 15, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154, 154; ...
MATRIX GENERATING METHOD.
Given triangle T(n, k) = binomial(k^3+n-k-1, n-k), such that the g.f. of column k equals 1/(1-x)^(k^3) for k>=1, which begins:
1;
1, 1;
1, 8, 1;
1, 36, 27, 1;
1, 120, 378, 64, 1;
1, 330, 3654, 2080, 125, 1;
1, 792, 27405, 45760, 7875, 216, 1;
1, 1716, 169911, 766480, 333375, 23436, 343, 1; ...
then this sequence forms column 1 (ignoring signs) of the matrix inverse:
1;
-1, 1;
7, -8, 1;
-154, 180, -27, 1;
7329, -8616, 1350, -64, 1;
-621054, 731502, -116244, 5920, -125, 1;
83287785, -98171784, 15685569, -820480, 19125, -216, 1;
-16339143828, 19265191212, -3085386984, 163253040, -3963750, 50652, -343, 1; ...

Cf. A177447.

/* GENERATING FUNCTION: */
{a(n)=local(F=1/(1+x+x*O(x^n))); polcoeff(x-sum(k=1, n-1, a(k)*x^k*F^(k^3)), n)}
for(n=1,20,print1(a(n),", "))

/* SUMMATION METHOD: */
{A=[1, 1]; for(i=1, 20, A=concat(A, -Vec(sum(n=0, #A-1, A[n+1]*x^n/(1+x+x*O(x^#A))^(n^3)))[#A+1])); for(n=1, #A-1, print1(A[n+1], ", "))}

/* MATRIX METHOD: */
{a(n)=local(M=matrix(n,n,r,c,if(r>=c,binomial(c^3+r-c-1, r-c))));-(-1)^n*(M^-1)[n,1]}
for(n=1,20,print1(a(n),", "))

A229806 G.f.: Sum_{n>=1} a(n)x^n / (1 + nx)^(n^2) = x.

Original entry on oeis.org

1, 1, 7, 150, 6924, 569726, 74358042, 14229990742, 3774315375580, 1330122245198910, 602741550311798067, 342138788139339603446, 238146938124253555981224, 199695655908033678248780110, 198741234873020798204357773510, 231773141251670398730627959107510

Offset: 1

Paul D. Hanna, Oct 02 2013

nonn

Compare to identity: Sum_{n>=1} n^(n-2) * x^n / (1 + n*x)^n = x.

			G.f.: x = 1*x/(1+x) + 1*x^2/(1+2*x)^4 + 7*x^3/(1+3*x)^9 + 150*x^4/(1+4*x)^16 + 6924*x^5/(1+5*x)^25 + 569726*x^6/(1+6*x)^36 +...

Cf. A177447, A082157.

{a(n)=polcoeff(x-sum(k=1, n-1, a(k)*x^k/(1+k*x+x*O(x^n))^(k^2)), n)}
for(n=1,20,print1(a(n),", "))

cp's OEIS Frontend

A133316 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n/(1+x)^(n^2).

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PARI

PARI

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A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

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Maple

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A215241 Unsigned matrix inverse of triangle A214398, as a triangle read by rows n >= 1.

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Mathematica

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A215242 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^((n+2)^2).

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PARI

A215243 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^((n+3)^2).

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PARI

A177448 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(2*n^2) = 1+x.

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PARI

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A177449 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(3*n^2) = 1+x.

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A229711 G.f.: Sum_{n>=0} a(n)*x^n / (1+x)^(n^3) = x.

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PARI

PARI

PARI

A229806 G.f.: Sum_{n>=1} a(n)*x^n / (1 + n*x)^(n^2) = x.

A177448 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(2n^2) = 1+x.

A177449 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(3n^2) = 1+x.

A229806 G.f.: Sum_{n>=1} a(n)x^n / (1 + nx)^(n^2) = x.