Real-rooted atlas

A143491 → basis A013611 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(8 + 4\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 56 + 60\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0566016132
1504764384641716
2504096566896217625624024
35544012637611448051520115201024360360
46652801738272187926410761603443205836840965765760
58648640252586563138352021507136878080021360642867201638498017920
612108096038821574454040390442663398420895974465028096125583361376256655361764322560
718162144006307560000965892153685611253764840932096181126041644848742470877184648806426214433522128640
82905943040010818581760017977298457617561369216011169941504048343895040144208404482927984640387317760301465601048576670442572800
949401031680019553966208003488884008192370452470502426013448243201268643875840438529867776107459100672182963404802061762560138412032419430414079294028800
10889218570240037173180441600706214986306568063698072320061642305657856332409690624001296811312332836883832832007591705313281102970880001073846681662914560016777216309744468633600
11168951528345600741859171200000149050119574886418145886282634241493751730392064878147634817024379358025592832121951734874112291777732280325132326797312645219221504549076336642835349504671088647124122778572800
1233790305669120001551298953738240032777460599777280422537773482639363713338912089497623537959617908736110997510511247363956466799853568107136240405708821935762885836833433691619328367902955929627633752473612683575296268435456170978946685747200
1370959641905152000339288902552678400750378630744852480101843916671265177694881628093185024064283070845966336032724661054525440012748480700142387238324477684613120889195982225408015795380394393602109943872225282051920625664013717051801605637144576010737418244274473667143680000
1415611121219133440007748194423779532800178654854865974681602540717619065774899224947714847351312384179375407098399948809770748265834250240411365219621234278413530777370671841283489210268280422407031767615668224010960028676653056129540008653619211225433899008067269925273602491081031684294967296111136315345735680000
Technical details
Node id
n-adf12b41f5815133 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-47b873245365cff6

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:           56,            60,            16;
  1:          504,           764,           384,            64;
  2:         5040,          9656,          6896,          2176,           256;
  3:        55440,        126376,        114480,         51520,         11520,          1024;
  4:       665280,       1738272,       1879264,       1076160,        344320,         58368,         4096;
  5:      8648640,      25258656,      31383520,      21507136,       8780800,       2136064,       286720,        16384;
  6:    121080960,     388215744,     540403904,     426633984,     208959744,      65028096,     12558336,      1376256,       65536;
  7:   1816214400,    6307560000,    9658921536,    8561125376,    4840932096,    1811260416,    448487424,     70877184,     6488064,     262144;
  8:  29059430400,  108185817600,  179772984576,  175613692160,  111699415040,   48343895040,  14420840448,   2927984640,   387317760,   30146560,   1048576;
  9: 494010316800, 1955396620800, 3488884008192, 3704524705024, 2601344824320, 1268643875840, 438529867776, 107459100672, 18296340480, 2061762560, 138412032, 4194304;
%F Formula
The row polynomials satisfy P(n,x) = (n+8+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 56 + 60*x + 16*x^2.
%t Mathematica
P[0, x_] := 56 + 60*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 8 + 4*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] = 56 + 60*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 8 + 4*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.