Real-rooted atlas

A038221 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,n x\right)\,P_{n}(x) = \left(6 + 24\,x + 24\,x^{2} + 12\,n x + 24\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
162430
236288432756
321625927776691217496
412962073693312165888103680384912
577761555209331202488320311040014929928188128
6466561119744839808029859840559872005374771220901888170061120
727993678382087054387231352832078382080011287019528778792962866544643469246848
816796165374771256435097630098718729405849600180592312322106910310413759414272386983526469793083648
910077696362797056435356467227088846848101583175680243799621632379243855872371504185344208971104256515978035201388515032576
106046617624186470403265173504023219011584010158317568002925595459584568865783808074300837068806269133127680309586821120068109100646427373582070784
1136279705615963070464239446056960191556845568095778422784003218155005542475090283462656122596381163520137920928808960102163650969600449520064266248916100448256535570083993600
12217678233610448555212817240116101121532454764544086200580505600331010229141504901083401551872176538788875468824825767185612802451927623270400161827223135846464195923227443211590930582732810411482432835584
131306069401667915608883212224809598976119531471634432747071697715200322734973412966410040643617292288229500425538109443872819680955596847812588653772800420750780153200642503641005870284890409258545315841497904875307008201288660368154624
1478364164096438839318937685573667192832912785783390208627540226080768030121930851876864105426757981569024275400510645731328542194755333783552803251489383383040883576638321721344701019481643679744379718885890326528125824009525788672192587769682329603873071465014837248
Technical details
Node id
n-c88e193760555c01 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7cabb77b45806852

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:        6,        24;
  2:       36,       288,        432;
  3:      216,      2592,       7776,        6912;
  4:     1296,     20736,      93312,      165888,       103680;
  5:     7776,    155520,     933120,     2488320,      3110400,      1492992;
  6:    46656,   1119744,    8398080,    29859840,     55987200,     53747712,     20901888;
  7:   279936,   7838208,   70543872,   313528320,    783820800,   1128701952,    877879296,    286654464;
  8:  1679616,  53747712,  564350976,  3009871872,   9405849600,  18059231232,  21069103104,  13759414272,   3869835264;
  9: 10077696, 362797056, 4353564672, 27088846848, 101583175680, 243799621632, 379243855872, 371504185344, 208971104256, 51597803520;
%F Formula
The row polynomials satisfy (1+2*n*x)*P(n,x) = (6+(12*n+24)*x+(24*n+24)*x^2)*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[Together[((6 + (12*n + 24)*x + (24*n + 24)*x^2)*P[n - 1, x])/(1 + 2*n*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] = ((6 + (12*n + 24)*x + (24*n + 24)*x^2)*v[i - 1])/(1 + 2*n*x)); 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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.