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The number of Feynman graphs in phi^4 theory with n vertices, 2 external legs.

2

`%I #29 Jun 10 2024 08:50:55
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`%S 1,2,7,23,85,340,1517,7489,41276,252410,1706071,12660012,102447112,
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`%T 898081422,8477941776,85729296020,924345402273,10584325318278,
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`%U 128259347448244,1639694094741643,22053783907891362,311294619360437722,4601020643330758040,71063337073204684379,1144820435086864897289
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`%N The number of Feynman graphs in phi^4 theory with n vertices, 2 external legs.
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`%C The generating function of this is the product of the g.f. of the connected diagrams (A352174) by the g.f. of the vacuum diagrams (A129429, including a term x^0 for the empty graph): x + 2*x^2 + 7*x^3 + 23*x^4 + ... = (x + x^2 + 3*x^3 + 10*x^4 + ...) * (1 + x + 3*x^2 + 7*x^3 + 20*x^4 + ...). - _R. J. Mathar_, Mar 05 2023
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`%C a(n) is the number of multigraphs with n unlabeled nodes of degree 4 plus 2 noninterchangeable nodes of degree 1, loops allowed. - _Andrew Howroyd_, Mar 10 2023
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`%H R. de Mello Koch and S. Ramgoolam, <a href="https://doi.org/10.1103/PhysRevD.85.026007">Strings from Feynman graph counting: Without large N</a>, Phys. Rev. D 85 (2012) 026007, App. D.
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`%Y Cf. A352174 (connected), A129429 (0 ext. legs), A352175 (degree 3 case).
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`%K nonn
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`%O 0,2
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`%A _R. J. Mathar_, Mar 07 2022
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`%E Offset corrected and a(13) and beyond from _Andrew Howroyd_, Mar 10 2023
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