A127059 -id:A127059 - cp's OEIS frontend

A167872 A sequence of moments connected with Feynman numbers (A000698): Half the number of Feynman diagrams of order 2(n+1), for the electron self-energy in quantum electrodynamics (QED), i.e., all proper diagrams including Furry vanishing diagrams (those that vanish in 4-dimensional QED because of Furry theorem).

1, 3, 21, 207, 2529, 36243, 591381, 10786527, 217179009, 4782674403, 114370025301, 2952426526767, 81864375589089, 2427523337157363, 76683680366193621, 2571609710380950207, 91265370849151405569, 3417956847888948899523

Offset: 0

Groux Roland, Nov 14 2009

nonn

a(n) is the moment of order 2*n of the probability density function defined by rho(x) = sqrt(Pi/2)*exp(-x^2/2)/((x*phi(x)+1)^2 + Pi^2*x^2*exp(-x^2)), where phi(x) = Integral_{t=-oo..oo} t*log(abs(x-t))*exp(-t^2/2) dt.

			G.f. = 1 + 3*x + 21*x^2 + 207*x^3 + 2529*x^4 + 36243*x^5 + 591381*x^6 + ...

Roland Groux. Polynômes orthogonaux et transformations intégrales. Cepadues. 2008. pages 195..206.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Trinh Khanh Duy and Tomoyuki Shirai, The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, arXiv preprint arXiv:1504.06904 [math.SP], 2015.
Adrian Ocneanu, On the inner structure of a permutation: bicolored partitions and Eulerians, trees and primitives; arXiv preprint arXiv:1304.1263 [math.CO], 2013.
Wikipedia, Feynman diagram

Row 3 of A321964. Cf. A000698, A115974, A005411, A005412, A001147.

(* f = A000698 *) f[n_] := f[n] = (2*n - 1)!! - Sum[f[n - k]*(2*k - 1)!!, {k, 1, n - 1}]; a[n_] := a[n] = f[n + 2]/2 - Sum[f[n + 1 - k]*a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 03 2013, from 3rd formula *)
nmax = 20; CoefficientList[Series[1/(1 + x + ContinuedFractionK[-(k - (-1)^k)*x, 1, {k, 3, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2022, after Peter Bala *)

{a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2*k - 3) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 23 2011 */

Sum_{n>=0} a(n)/z^(2n+1) = (1/2)*(z-S(z)/(z*S(z)-1)) with S(z) = Sum_{n>=0} (2*n)!/(2^n*n!*z^(2*n+1)).
a(n) = (2*n - 1) * a(n-1) + 2 * Sum_{k=1..n} a(k-1) * a(n-k) if n>0. - Michael Somos, Jul 23 2011
a(0)=1; for n > 0, a(n) = A000698(n+2)/2 - Sum_{k=0..n-1} A000698(n+1-k)*a(k).
G.f.: 1/(1-3*x/(1-4*x/(1-5*x/(1-6*x/(1-7*x/(1-8*x/(...))))))) (continued fraction). - Philippe Deléham, Nov 20 2011
G.f.: 1/Q(0), where Q(k) = 1 - x*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
Let A(x) be the g.f. of A127059 and B(x) be the g.f. of A167872. Then A(x) = (1 - 1/B(x))/x.
G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k+3)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - (2*k+3)*x/((2*k+2)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
G.f.: G(0), where G(k) = 1 - x*(k+3)/(x*(k+3) - 1/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Aug 05 2013
a(n) = A115974(n)/2, see comments in A115974. See also A000698, A005411, A005412. - Robert Coquereaux, Sep 14 2014
a(n) ~ 2^(n + 3/2) * n^(n+2) / exp(n). - Vaclav Kotesovec, Jan 02 2019
G.f.: 1/(1 + x - 4*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 8*x/(1 - 7*x/(1 - ...))))))). - Peter Bala, May 30 2022

Name clarified from Robert Coquereaux, Sep 14 2014

A127058 Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.

Original entry on oeis.org

1, 2, 2, 10, 6, 3, 74, 42, 12, 4, 706, 414, 108, 20, 5, 8162, 5058, 1332, 220, 30, 6, 110410, 72486, 19908, 3260, 390, 42, 7, 1708394, 1182762, 342252, 57700, 6750, 630, 56, 8, 29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9, 576037442

Offset: 0

Paul D. Hanna, Jan 04 2007

Column 0 is A000698, the number of shellings of an n-cube, divided by 2^n n!.

Column 1 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.

			Other recurrences exist, as shown by:
column 0 = A000698: T(n,0) = (2n+1)!! - Sum_{k=1..n} (2k-1)!!*T(n-k,0);
column 1 = A115974: T(n,1) = T(n+1,0) - Sum_{k=0..n-1} T(k,1)*T(n-k,0).
Illustrate the recurrence:
T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) (n > k >= 0)
at column k=1:
T(2,1) = T(1,1)*T(2,2) = 2*3 = 6;
T(3,1) = T(1,1)*T(3,2) + T(2,1)*T(2,2) = 2*12 + 6*3 = 42;
T(4,1) = T(1,1)*T(4,2) + T(2,1)*T(3,2) + T(3,1)*T(2,2) = 2*108 + 6*12 + 42*3 = 414;
at column k=2:
T(3,2) = T(2,2)*T(3,3) = 3*4 = 12;
T(4,2) = T(2,2)*T(4,3) + T(3,2)*T(3,3) = 3*20 + 12*4 = 108;
T(5,2) = T(2,2)*T(5,3) + T(3,2)*T(4,3) + T(4,2)*T(3,3) = 3*220 + 12*20 + 108*4 = 1332.
Triangle begins:
         1;
         2,        2;
        10,        6,       3;
        74,       42,      12,       4;
       706,      414,     108,      20,      5;
      8162,     5058,    1332,     220,     30,     6;
    110410,    72486,   19908,    3260,    390,    42,   7;
   1708394,  1182762,  342252,   57700,   6750,   630,  56,  8;
  29752066, 21573054, 6583788, 1159700, 138150, 12474, 952, 72, 9; ...

G. C. Greubel, Rows n = 0..15 of triangle, flattened

Columns: A000698, A115974, A127059.
Row sums: A127060.
Cf. A001147 ((2n-1)!!).

T[n_,k_]:= If[k==n, n+1, Sum[T[j+k,k]*T[n-j,k+1], {j,0,n-k-1}]];
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 03 2019 *)

{T(n,k)=if(n==k,n+1,sum(j=0,n-k-1,T(j+k,k)*T(n-j,k+1)))}

def T(n, k):
    if (k==n): return n+1
    else: return sum(T(j+k,k)*T(n-j,k+1) for j in (0..n-k-1))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 03 2019

A127060 Row sums of triangle A127058.

Original entry on oeis.org

1, 4, 19, 132, 1253, 14808, 206503, 3298552, 59220265, 1179047100, 25767347387, 613141219356, 15780105110605, 436801028784112, 12941788708753999, 408718346076189360, 13707898517284016849, 486640514520848512692

Offset: 0

Paul D. Hanna, Jan 04 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A127058, A127059.

T[n_, k_]:=T[n, k]=If[k==n, n+1, Sum[T[j+k,k]*T[n-j,k+1], {j,0,n-k-1}]];
Table[Sum[T[n, j], {j,0,n}], {n,0,20}] (* G. C. Greubel, Jun 08 2019 *)

getT(n, k, T) = if (!T[n+1,k+1], T[n+1, k+1] = sum(j=0, n-k-1, getT(j+k, k, T)*getT(n-j, k+1, T))); T[n+1, k+1];
tabl(nn) = {my(T = matrix(nn+1, nn+1)); for (i=1, nn+1, T[i, i] = i); for (i=0, nn, for (j=0, i, T[i+1, j+1] = getT(i, j, T); ); ); T; } /* A127059 */
lista(nn) = {my(T = tabl(nn)); vector(nn, k, vecsum(T[k, ]));}
lista(20) \\ Michel Marcus, Jun 09 2019

@CachedFunction
def T(n, k):
    if (k==n): return n+1
    else: return sum(T(j+k, k)*T(n-j, k+1) for j in (0..n-k-1))
def a(n): return sum(T(n,j) for j in (0..n))
[a(n) for n in (0..20)] # G. C. Greubel, Jun 08 2019

a(17) corrected by G. C. Greubel, Jun 08 2019

A321963 Stieltjes generated from the sequence m, m+1, m+2, m+3, .... where m = 4.

Original entry on oeis.org

1, 4, 36, 444, 6636, 114084, 2194596, 46460124, 1070653356, 26650132164, 712373143716, 20355134459004, 619356569885676, 20002325474150244, 683641504802995236, 24662695086736585884, 936845038595867508396, 37388655553571504769924, 1564425694139017014501156

Offset: 0

Peter Luschny, Dec 26 2018

nonn

See A321964 for the definitions.

A000007 (m=0), A001147 (m=1), A000698 (m=2), A167872 (m=3), this sequence (m=4).

a(n) = A127059(n)/3.

A321963List := proc(len) local S, k, m, cf, ser;
    S := [seq(k+4, k = 0..len)]: m := 1;
    for k from len by -1 to 1 do
        m := 1 - S[k]*x/m od;
    cf := 1/m:
    ser := series(cf, x, len);
    seq(coeff(ser, x, n), n = 0..len-1) end:
A321963List(19);

T[n_, k_] := T[n, k] = If[k == n, n + 1, Sum[T[j + k, k] T[n - j, k + 1], {j, 0, n - k - 1}]]; a[n_] := T[n + 2, 2]/3; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 22 2019, from A127059 *)

a(n) ~ 2^(n + 5/2) * n^(n+3) / (3*exp(n)). - Vaclav Kotesovec, Jan 02 2019

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A127058 Triangle, read by rows, defined by: T(n,k) = Sum_{j=0..n-k-1} T(j+k,k)*T(n-j,k+1) for n > k >= 0, with T(n,n) = n+1.

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A127060 Row sums of triangle A127058.

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A321963 Stieltjes generated from the sequence m, m+1, m+2, m+3, .... where m = 4.

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