Real-rooted atlas

A094638 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(x + 2\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2814628
3481169224280
4384116813166521203640
53840139842016814416511272058240
6460801946883399043141681622564442450401106560
7645120309427263161607117584478492819199844259524032024344320
8103219205531443212890700817072678414061710473784096240950884478688362880608608000
918579456010988789762873470464436215219242383757122734284768117155254432156726451318720362880017041024000
10371589120024021319680695570780161188512189441327511883521013078281925350818332819318423264456361430463708192039916800528271744000
118174960640057305972736018185115525123449411752960434674077107238217864804482392873971520106710351174433222059385668779173888852315264047900160017961239296000
12196199055360014816178339840510940537159681064265322536961491641312942081482305055546881071121995623045671784591161621845639905216596956789344010986849239041222970227206227020800664565853952000
135101175439360041268850458624015358718933729283482406590619648536823886520115259422909825408004860144266385536297423478757427213620364802982404610477239024641121397585296641856131152537618740608588808717829120026582634158080000
141428329123020800123204544443187204919474058323558412054546293794406420254678708492697624690773048915968022521840419690700815618073804786265682750543261964800333398834737425921005562909736755222018131006554883308933769292803055190503680013076743680001143053268797440000
Technical details
Node id
n-d44f94dfc4c66d6b · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-2f2b2d9b62c79a93

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         2,          2;
  2:         8,         14,          6;
  3:        48,        116,         92,         24;
  4:       384,       1168,       1316,        652,        120;
  5:      3840,      13984,      20168,      14416,       5112,        720;
  6:     46080,     194688,     339904,     314168,     162256,      44424,       5040;
  7:    645120,    3094272,    6316160,    7117584,    4784928,    1919984,     425952,     40320;
  8:  10321920,   55314432,  128907008,  170726784,  140617104,   73784096,   24095088,   4478688,   362880;
  9: 185794560, 1098878976, 2873470464, 4362152192, 4238375712, 2734284768, 1171552544, 321567264, 51318720, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+(n+1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n + (n + 1)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (2*n + (n + 1)*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.