A321964 -id:A321964 - cp's OEIS frontend

A321961 Sums of antidiagonals of A321964.

1, 1, 2, 6, 29, 205, 1900, 21592, 289325, 4456773, 77512394, 1501537750, 32053808641, 747561173689, 18911411905052, 515798244940224, 15088848988966409, 471291927943037785, 15655172277281565058, 551102758946418279862, 20495154400760992784213, 802951447375696586076213

Offset: 0

Peter Luschny, Dec 27 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A321964.

def A321961List(l):
    def row(n, dim):
        m = 1
        for k in range(dim-1, -1, -1):
            m = 1 - (n+k)*x/m
        return SR(1/m).series(x, dim).list()
    return [sum(row(n-k, n+1)[k] for k in (0..n)) for n in (0..l-1)]
A321961List(16)

a(n) ~ 2^(n - 1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Jan 06 2019

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).

Original entry on oeis.org

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

A321960 Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, ....

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 92, 57, 20, 5, 1, 0, 203, 426, 303, 116, 30, 6, 1, 0, 877, 2146, 1752, 744, 205, 42, 7, 1, 0, 4140, 11624, 10845, 5140, 1535, 330, 56, 8, 1, 0, 21147, 67146, 71139, 37676, 12300, 2820, 497, 72, 9, 1

Offset: 0

Peter Luschny, Dec 27 2018

For definitions and comments see A321964.

			First few rows of the array start:
[0] 1, 0,  0,   0,    0,     0,      0,       0,        0, ... A000007
[1] 1, 1,  2,   5,   15,    52,    203,     877,     4140, ... A000110
[2] 1, 2,  6,  22,   92,   426,   2146,   11624,    67146, ... A074664
[3] 1, 3, 12,  57,  303,  1752,  10845,   71139,   491064, ... A321959
[4] 1, 4, 20, 116,  744,  5140,  37676,  290224,  2334300, ...
[5] 1, 5, 30, 205, 1535, 12300, 103975,  918785,  8434740, ...
[6] 1, 6, 42, 330, 2820, 25662, 245358, 2443272, 25188870, ...
[7] 1, 7, 56, 497, 4767, 48496, 516761, 5719399, 65369136, ...
Seen as triangle:
[0] 1;
[1] 0,   1;
[2] 0,   1,    1;
[3] 0,   2,    2,    1;
[4] 0,   5,    6,    3,   1;
[5] 0,  15,   22,   12,   4,   1;
[6] 0,  52,   92,   57,  20,   5,  1;
[7] 0, 203,  426,  303, 116,  30,  6, 1;
[8] 0, 877, 2146, 1752, 744, 205, 42, 7, 1;

Rows of array: A000007, A000110, A074664, A321959.
Columns include: A002378, A033445. Row sums of triangle: A321958.
Cf. A321964.

# The function JacobiSquare is defined in A321964.
s := n -> [seq(n+k, k = 0..9)]: Trow := n -> JacobiSquare(s(n)):
for n from 0 to 7 do lprint(Trow(n)) od;

nmax = 10;
JacobiCF[a_, b_, p_:2] := Module[{m, k}, m = 1; For[k = Length[a], k >= 1, k--, m = 1 - b[[k]]*x - a[[k]]*x^p/m]; 1/m];
JacobiSquare[a_, p_: 2] := Module[{cf, ser}, cf = JacobiCF[a, a, p]; ser = Series[cf, {x, 0, Length[a]}]; CoefficientList[ser, x]];
s[n_] := Table[n + k, {k, 0, nmax}];
row[n_] := row[n] = JacobiSquare[s[n]];
T[, 0] = 1; T[0, ] = 0; T[n_, k_] := row[n][[k + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 13 2019, after Peter Luschny in A321964 *)

def JacobiCF(a, b, dim, p=2):
    m = 1
    for k in range(dim-1, -1, -1):
        m = 1 - b(k)*x - a(k)*x^p/m
    return 1/m
def JacobiGF(a, b, dim, p=2):
    cf = JacobiCF(a, b, dim, p)
    return cf.series(x, dim).list()
def JacobiSquare(a, dim, p=2):
    cf = JacobiCF(a, a, dim, p)
    return cf.series(x, dim).list()
def StieltjesGF(a, dim, p=2):
    return JacobiGF(a, lambda n: 0, dim, p)
def Trow(n): return JacobiSquare(lambda k: n+k, 10)
for n in (0..4): print(Trow(n))

T(n, k) = A(n)[k] where A(n) is the Jacobi square of the sequence s(j) = n + j, j >= 0.

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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A321961 Sums of antidiagonals of A321964.

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A321960 Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, ....

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Author

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Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

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

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Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula