A117402 -id:A117402 - cp's OEIS frontend

A117401 Triangle T(n,k) = 2^(k*(n-k)), read by rows.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 8, 1, 1, 16, 64, 64, 16, 1, 1, 32, 256, 512, 256, 32, 1, 1, 64, 1024, 4096, 4096, 1024, 64, 1, 1, 128, 4096, 32768, 65536, 32768, 4096, 128, 1, 1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1

Offset: 0

Paul D. Hanna, Mar 12 2006

Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character.

			A(x,y) = 1/(1-xy) + x/(1-2xy) + x^2/(1-4xy) + x^3/(1-8xy) + ...
Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   4,     4,      1;
  1,   8,    16,      8,       1;
  1,  16,    64,     64,      16,       1;
  1,  32,   256,    512,     256,      32,      1;
  1,  64,  1024,   4096,    4096,    1024,     64,     1;
  1, 128,  4096,  32768,   65536,   32768,   4096,   128,   1;
  1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A117402 (row sums), A117403 (antidiagonal sums), A002416 (central terms).

Cf. this sequence (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).

A117401:= func< n, k, m | (m+2)^(k*(n-k)) >;
[A117401(n, k, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

Table[2^((n-k)k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 09 2017 *)

PARI
```
T(n,k)=if(n
				
```

def A117401(n, k, m): return (m+2)^(k*(n-k))
flatten([[A117401(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

G.f.: A(x,y) = Sum_{n>=0} x^n/(1 - 2^n*x*y).
G.f. satisfies: A(x,y) = 1/(1 - x*y) + x*A(x,2*y).
Equals ConvOffsStoT transform of the 2^n series: (1, 2, 4, 8, ...); e.g., ConvOffs transform of (1, 2, 4, 8) = (1, 8, 16, 8, 1). - Gary W. Adamson, Apr 21 2008
T(n,k) = (1/n)*( 2^(n-k)*k*T(n-1,k-1) + 2^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
Let E(x) = Sum_{n>=0} x^n/2^C(n,2).  Then E(x)*E(y*x) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/2^C(n,2). - Geoffrey Critzer, May 31 2020
T(n, k, m) = (m+2)^(k*(n-k)) with m = 0. - G. C. Greubel, Jun 28 2021

A349893 a(n) = Sum_{k=0..n} k^(k*(n-k)).

Original entry on oeis.org

1, 2, 3, 7, 46, 1052, 88603, 27121965, 37004504306, 198705527223758, 5595513387083114571, 686714367475480207331583, 468422339816915120237104999422, 1664212116512828935888786624225704856, 31295654819650678010096952493864470025103251

Offset: 0

Seiichi Manyama, Dec 04 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..52

Cf. A117402, A349880, A349881, A349886, A349894.

Table[1 + Sum[k^(k*(n - k)), {k, 1, n}], {n, 0, 16}] (* Vaclav Kotesovec, Dec 05 2021 *)

PARI
```
a(n) = sum(k=0, n, k^(k*(n-k)));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^k*x)))

G.f.: Sum_{k>=0} x^k/(1 - k^k * x).

log(a(n)) ~ n^2*log(n)/4 * (1 - log(2)/log(n) + 1/(4*log(n)^2)). - Vaclav Kotesovec, Dec 05 2021

A117403 a(n) = Sum_{k=0..floor(n/2)} 2^((n-2k)k) for n>=0.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 34, 105, 386, 1681, 8706, 53793, 395266, 3442753, 35659778, 440672385, 6476038146, 112812130561, 2336999211010, 57759810847233, 1697654543745026, 59146046307566593, 2450521284684021762

Offset: 0

Paul D. Hanna, Mar 12 2006

nonn

Equals the antidiagonal sums of triangle A117401.

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 13*x^5 + 34*x^6 + 105*x^7 + ...
where
A(x) = 1/(1-x^2) + x/(1-2*x^2) + x^2/(1-4*x^2) + x^3/(1-8*x^2) + x^4/(1-16*x^2) + ...

G. C. Greubel, Table of n, a(n) for n = 0..150

Cf. A117401 (triangle), A117402 (row sums).

[(&+[2^(k*(n-2*k)) : k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 28 2021

Table[Sum[2^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Jun 28 2021 *)

PARI
```
a(n) = sum(k=0,n\2,2^((n-2*k)*k))
```

{a(n) = polcoeff(sum(m=0,n,x^m/(1-2^m*x^2 +x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

[sum(2^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # G. C. Greubel, Jun 28 2021

G.f.: A(x) = Sum_{n>=0} x^n / (1 - 2^n*x^2).
a(2*n) = Sum_{k=0..n} 4^((n-k)*k).
a(2*n+1) = Sum_{k=0..n} 2^k * 4^((n-k)*k).
G.f.: 1/(1-x^2) - x/(Q(0) +x-x^3), where Q(k) = x^2*(2+x)*2^k -1-x - x*(2*x^2*2^k -1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 11 2013

Name changed by Paul D. Hanna, Nov 11 2013

A360704 Expansion of Sum_{k>=0} (x * (1 + 2^k * x))^k.

Original entry on oeis.org

1, 1, 3, 9, 41, 257, 2209, 27009, 455553, 10831873, 360452609, 16786663425, 1102243190785, 101146710556673, 13109796072955905, 2379217548538511361, 609386444958743363585, 219178211386515281412097, 111098724276069341895720961, 79284929294467154275606200321

Offset: 0

Seiichi Manyama, Feb 17 2023

nonn

Cf. A117402, A360592, A360696, A360705.

Cf. A000684.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+2^k*x))^k))

a(n) = sum(k=0, n\2, 2^(k*(n-k))*binomial(n-k, k));

a(n) = Sum_{k=0..floor(n/2)} 2^(k*(n-k)) * binomial(n-k,k).

A337161 Square array read by antidiagonals: T(n,k) is the number of simple labeled graphs G with vertex set V(G) = {v_1,...,v_n} along with a (coloring) function C:V(G) ->[k] such that v_i adjacent to v_j implies C(v_i) != C(v_j) and i=0, k>=0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 10, 1, 0, 1, 5, 16, 35, 34, 1, 0, 1, 6, 25, 84, 195, 162, 1, 0, 1, 7, 36, 165, 644, 1635, 1090, 1, 0, 1, 8, 49, 286, 1605, 7620, 21187, 10370, 1, 0, 1, 9, 64, 455, 3366, 24389, 143748, 430467, 139522, 1, 0, 1, 10, 81, 680, 6279, 62310, 599685, 4412164, 13812483, 2654722, 1, 0, 1, 11, 100, 969, 10760, 136871, 1882054, 24413445, 223233540, 702219779, 71435266, 1, 0

Offset: 0

Geoffrey Critzer, Jan 28 2021

			  1, 1,    1,     1,      1,      1,       1, ...
  0, 1,    2,     3,      4,      5,       6, ...
  0, 1,    4,     9,     16,     25,      36, ...
  0, 1,   10,    35,     84,    165,     286, ...
  0, 1,   34,   195,    644,   1605,    3366, ...
  0, 1,  162,  1635,   7620,  24389,   62310, ...
  0, 1, 1090, 21187, 143748, 599685, 1882054, ...

R. P. Stanley, Enumerative Combinatorics, Vol I, Second Edition, Section 3.18.

Cf. A322280, A117402 (column k=2).

nn = 6; e[x_] := Sum[x^n/(2^Binomial[n, 2]), {n, 0, nn}];
Table[Table[2^Binomial[n, 2], {n, 0, nn}] PadRight[CoefficientList[Series[e[x]^k, {x, 0, nn}], x], nn + 1], {k, 0, nn}] // Transpose // Grid

Let e(x) = Sum_{n>=0} x^n/2^binomial(n,2). Then e(x)^k = Sum_{n>=0} Z_n(k)*x^n/2^biomial(n,2) and T(n,k) = Z_n(k). Z_n(k) is the zeta polynomial of the class of posets described in A117402.

A341957 E.g.f. A(x) satisfies: Sum_{n>=0} A(x)^n * exp(2^nA(x)) / n! = Sum_{n>=0} x^n/(1 - 2^nx).

Original entry on oeis.org

1, 1, 8, 144, 4554, 230940, 18177900, 2196712980, 406693854000, 115319921466960, 50017977456121080, 33099984846144881280, 33309128229289401091680, 50790831819884758635873840, 116936359809482874588941613600, 405126119455062475269210516705600

Offset: 1

Paul D. Hanna, Mar 09 2021

nonn

			E.g.f.: A(x) = x + x^2/2! + 8*x^3/3! + 144*x^4/4! + 4554*x^5/5! + 230940*x^6/6! + 18177900*x^7/7! + 2196712980*x^8/8! + 406693854000*x^9/9! + 115319921466960*x^10/10! + ...
such that
Sum_{n>=0} A(x)^n * exp(2^n*A(x)) / n! = exp(A(x)) + A(x)*exp(2*A(x)) + A(x)^2*exp(2^2*A(x))/2! + A(x)^3*exp(2^3*A(x))/3! + A(x)^4*exp(2^4*A(x))/4! +...
equals the sum
Sum_{n>=0} x^n/(1 - 2^n*x) = 1 + 2*x + 4*x^2 + 10*x^3 + 34*x^4 + 162*x^5 + 1090*x^6 + 10370*x^7 + 139522*x^8 + ... + A117402(n)*x^n + ...
RELATED SERIES.
exp(A(x)) = 1 + x + 2*x^2/2! + 12*x^3/3! + 186*x^4/4! + 5460*x^5/5! + 263940*x^6/6! + 20053740*x^7/7! + 2359326480*x^8/8! + 428122913400*x^9/9! + ...

Cf. A117402.

{a(n) = my(L=[0,1]); for(i=1,n, L=concat(L,0);
L[#L] = polcoeff( sum(n=0,#L, x^n/(1 - 2^n*x +x*O(x^#L))) - sum(n=0,#L, Ser(L)^n/n! * exp(2^n*Ser(L)) ) ,#L-1)/2;); n!*L[n+1]}
for(n=1,20,print1(a(n),", "))

cp's OEIS Frontend

A117401 Triangle T(n,k) = 2^(k*(n-k)), read by rows.

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A349893 a(n) = Sum_{k=0..n} k^(k*(n-k)).

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A117403 a(n) = Sum_{k=0..floor(n/2)} 2^((n-2*k)*k) for n>=0.

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A360704 Expansion of Sum_{k>=0} (x * (1 + 2^k * x))^k.

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PARI

PARI

Formula

A337161 Square array read by antidiagonals: T(n,k) is the number of simple labeled graphs G with vertex set V(G) = {v_1,...,v_n} along with a (coloring) function C:V(G) ->[k] such that v_i adjacent to v_j implies C(v_i) != C(v_j) and i=0, k>=0.

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Examples

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Crossrefs

Programs

Mathematica

Formula

A341957 E.g.f. A(x) satisfies: Sum_{n>=0} A(x)^n * exp(2^n*A(x)) / n! = Sum_{n>=0} x^n/(1 - 2^n*x).

Views

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Examples

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PARI

A117403 a(n) = Sum_{k=0..floor(n/2)} 2^((n-2k)k) for n>=0.

A341957 E.g.f. A(x) satisfies: Sum_{n>=0} A(x)^n * exp(2^nA(x)) / n! = Sum_{n>=0} x^n/(1 - 2^nx).