A178325 -id:A178325 - cp's OEIS frontend

A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

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, 1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1, 1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1, 1, 220, 12870, 170544

Offset: 1

Paul D. Hanna, Jul 15 2012

This is also the array A(n,k) read upwards antidiagonals, where the entry in row n and column k counts the vertex-labeled digraphs with n arcs and k vertices, allowing multi-edges and multi-loops (labeled analog to A138107). The binomial formula counts the weak compositions of distributing n arcs over the k^2 positions in the adjacency matrix. - R. J. Mathar, Aug 03 2017

			Triangle begins:
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;
1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1;
1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1;
1, 220, 12870, 170544, 593775, 658008, 270725, 45760, 3321, 100, 1; ...

Paul D. Hanna, Rows n = 0..45, flattened.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 80.

Cf. A214400 (central terms), A178325 (row sums), A054688, A000290 (1st subdiagonal), A037270 (2nd subdiagonal).

Cf. A230049.

A214398 := proc(n,k)
    binomial(k^2+n-k-1,n-k) ;
end proc:
seq(seq(A214398(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 03 2017

nmax = 11;
T[n_, k_] := SeriesCoefficient[1/(1-x)^(k^2), {x, 0, n-k}];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten

T(n,k)=binomial(k^2+n-k-1,n-k)
for(n=1,11,for(k=1,n,print1(T(n,k),", "));print(""))

T(n,k) = binomial(k^2+n-k-1, n-k).
Row sums form A178325.
Central terms form A214400.
T(n,n-2) = A037270(n-2). - R. J. Mathar, Aug 03 2017
T(n,n-3) = (n^2-6*n+11)*(n^2-6*n+10)*(n-3)^2 /6. - R. J. Mathar, Aug 03 2017

A230050 G.f.: Sum_{n>=0} x^n / (1-x)^(n^3).

Original entry on oeis.org

1, 1, 2, 10, 65, 564, 6191, 82050, 1295263, 23764278, 499547080, 11892550569, 317112508944, 9392408105655, 306739296397827, 10973970687363844, 427724034697254939, 18073023112616933860, 824247511186225346295, 40415810147764633887442, 2123162727678797736474583

Offset: 0

Paul D. Hanna, Oct 06 2013

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 65*x^4 + 564*x^5 + 6191*x^6 + 82050*x^7 +...
where
A(x) = 1 + x/(1-x) + x^2/(1-x)^8 + x^3/(1-x)^27 + x^4/(1-x)^64 + x^5/(1-x)^125 + x^6/(1-x)^216 + x^7/(1-x)^343 +...

Cf. A230049, A178325, A227934, A227935.

{a(n)=polcoeff(sum(k=0,n,x^k/(1-x+x*O(x^n))^(k^3)),n)}
for(n=0,25,print1(a(n),", "))

{a(n)=sum(k=0,n,binomial(k^3+n-k-1, n-k))}
for(n=0,25,print1(a(n),", "))

a(n) = Sum_{k=0..n} binomial(k^3 + n-k-1, n-k).

Equals row sums of triangle A230049.

A227934 G.f.: Sum_{n>=0} x^n / (1-x)^(n^4).

Original entry on oeis.org

1, 1, 2, 18, 219, 4395, 129280, 4970984, 257765641, 16781325293, 1348125117404, 132465548869248, 15490711962965785, 2134540479514352751, 343307151209151099650, 63606662918084631874716, 13470938654397531939066909, 3238387688528230753569245297, 876825599524773154743990986391

Offset: 0

Paul D. Hanna, Oct 06 2013

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 18*x^3 + 219*x^4 + 4395*x^5 + 129280*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/(1-x)^16 + x^3/(1-x)^81 + x^4/(1-x)^256 + x^5/(1-x)^625 + x^6/(1-x)^1296 + x^7/(1-x)^2401 +...

Cf. A178325, A230050, A227935.

{a(n)=polcoeff(sum(k=0,n,x^k/(1-x+x*O(x^n))^(k^4)),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=sum(k=0,n,binomial(k^4+n-k-1, n-k))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} binomial(k^4 + n-k-1, n-k).

A227935 G.f.: Sum_{n>=0} x^n / (1-x)^(n^5).

Original entry on oeis.org

1, 1, 2, 34, 773, 36656, 3001377, 333647780, 58561139773, 13838291852092, 4280413527001849, 1779704699369214238, 931039792575220097699, 604786686422678514970170, 489307443863919174036440087, 478922652139578822529676247092, 560120417434857039499787289137249

Offset: 0

Paul D. Hanna, Oct 06 2013

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 34*x^3 + 773*x^4 + 36656*x^5 + 3001377*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/(1-x)^32 + x^3/(1-x)^243 + x^4/(1-x)^1024 + x^5/(1-x)^3125 + x^6/(1-x)^7776 +...

Cf. A178325, A230050, A227934.

{a(n)=polcoeff(sum(k=0,n,x^k/(1-x+x*O(x^n))^(k^5)),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=sum(k=0,n,binomial(k^5+n-k-1, n-k))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} binomial(k^5 + n-k-1, n-k).

A325294 G.f. A(x) satisfies: Sum_{n>=0} x^nA(x)^(n(n+1)/2) = Sum_{n>=0} x^n/(1-x)^(n^2).

Original entry on oeis.org

1, 1, 2, 5, 17, 73, 368, 2074, 12663, 82236, 561664, 4004815, 29662508, 227413816, 1800063339, 14681764890, 123207630130, 1062547709801, 9407762681632, 85445941932906, 795514580068247, 7587015660017106, 74078917658328970, 740060483734580171, 7560421405484047766, 78939580213645975075, 841942979579094942598, 9168184497787176646141, 101876790751549107815492

Offset: 0

Paul D. Hanna, Apr 25 2019

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 73*x^5 + 368*x^6 + 2074*x^7 + 12663*x^8 + 82236*x^9 + 561664*x^10 + 4004815*x^11 + 29662508*x^12 + ...
such that the following series are equal:
B(x) = 1 + x*A(x) + x^2*A(x)^3 + x^3*A(x)^6 + x^4*A(x)^10 + x^5*A(x)^15 + x^6*A(x)^21 + x^7*A(x)^28 + x^8*A(x)^36 + ...
B(x) = 1 + x/(1-x) + x^2/(1-x)^4 + x^3/(1-x)^9 + x^4/(1-x)^16 + x^5/(1-x)^25 + x^6/(1-x)^36 + x^7/(1-x)^49 + x^8/(1-x)^64 + ...
where
B(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 83*x^5 + 363*x^6 + 1730*x^7 + 8889*x^8 + 48829*x^9 + 284858*x^10 + 1755325*x^11 + ... + A178325(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A178325, A325289, A326423.

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A, x^m*( Ser(A)^(m*(m+1)/2) - 1/(1-x +x*O(x^#A))^(m^2)) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

A214400 a(n) = binomial(n^2 + 3*n, n).

Original entry on oeis.org

1, 4, 45, 816, 20475, 658008, 25827165, 1198774720, 64276915527, 3911395881900, 266401260897200, 20082459351180240, 1660305826125766950, 149389005978091284720, 14533945899753270066525, 1520398315196482557890304, 170190601112537814791748255

Offset: 0

Paul D. Hanna, Jul 15 2012

nonn

Equals the central terms of triangle A214398.

Robert Israel, Table of n, a(n) for n = 0..336

Cf. A054688, A178325, A214398.

seq(binomial(n^2+3*n,n),n=0..30); # Robert Israel, Mar 04 2022

PARI
```
a(n)=binomial(n^2+3*n, n)
```

a(n) = [x^n] 1/(1 - x)^((n+1)^2). - Ilya Gutkovskiy, Oct 04 2017

a(n) ~ n^(n-1/2)*exp(n+5/2)/sqrt(2*Pi). - Robert Israel, Mar 04 2022

A214403 Triangle, read by rows of terms T(n,k) for k=0..n^2, that starts with a '1' in row 0 with row n>0 consisting of 2*n-1 '1's followed by the partial sums of the prior row.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 36, 47, 62, 83, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 19, 24, 30, 37, 45, 55, 68, 85, 107, 135, 171, 218, 280, 363

Offset: 0

Paul D. Hanna, Jul 15 2012

Right border and row sums form A178325.

			Triangle begins:
  [1];
  [1, 1];
  [1,1,1, 1, 2];
  [1,1,1,1,1, 1,2,3, 4, 6];
  [1,1,1,1,1,1,1, 1,2,3,4,5, 6,8,11, 15, 21];
  [1,1,1,1,1,1,1,1,1, 1,2,3,4,5,6,7, 8,10,13,17,22, 28,36,47, 62, 83];
  ...
Row sums equal the row sums (A178325) of triangle A214398,
where A214398(n, k) = binomial(k^2+n-k-1, n-k):
  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;
  ...

Paul D. Hanna, Rows n = 0..14, flattened.

Cf. A131338, A178325, A214398.

{T(n, k)=if(k>n^2||n<0||k<0, 0, if(n==0,1,if(k<=2*n-1, 1, sum(i=0, k-2*n+1, T(n-1, i)))))}
for(n=0,10,for(k=0,n^2,print1(T(n,k),", "));print(""))

A178323 Numbers n such that phi(reversal(n)) + sigma(reversal(n)) = n.

Original entry on oeis.org

572, 592, 5992, 599992, 2014080, 5999992, 594637872, 599999992, 599999999992

Offset: 1

Farideh Firoozbakht, May 28 2010

If n is in the sequence A070272 then reversal(n) is in this sequence. 10 divides all other terms of the sequence. 2014080 is the only known such term.
If p=6*10^n-1 is a prime greater than 5 then reversal(5*p) is in the sequence, see comment lines of A070272.
There is no further term up to 10^9.
10^12 < a(10) <= 1442827967760. - Giovanni Resta, Sep 04 2018

			2014080 = phi(804102) + sigma(804102), so 2014080 is in the sequence.

Cf. A000010, A000203, A004086, A070272, A178324, A178325.

r[n_]:=FromDigits[Reverse[IntegerDigits[n]]];
Do[If[EulerPhi[r[n]]+DivisorSigma[1,r[n]]==n,Print[n]],{n,1000000000}]

a(9) from Giovanni Resta, Sep 04 2018

cp's OEIS Frontend

A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

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A230050 G.f.: Sum_{n>=0} x^n / (1-x)^(n^3).

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A227934 G.f.: Sum_{n>=0} x^n / (1-x)^(n^4).

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A227935 G.f.: Sum_{n>=0} x^n / (1-x)^(n^5).

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A325294 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(n*(n+1)/2) = Sum_{n>=0} x^n/(1-x)^(n^2).

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A214400 a(n) = binomial(n^2 + 3*n, n).

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A214403 Triangle, read by rows of terms T(n,k) for k=0..n^2, that starts with a '1' in row 0 with row n>0 consisting of 2*n-1 '1's followed by the partial sums of the prior row.

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PARI

A178323 Numbers n such that phi(reversal(n)) + sigma(reversal(n)) = n.

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Comments

Examples

A325294 G.f. A(x) satisfies: Sum_{n>=0} x^nA(x)^(n(n+1)/2) = Sum_{n>=0} x^n/(1-x)^(n^2).