Real-rooted atlas

A147716 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
119423
236115216529
3685943329126412167
4130321109744346564864256279841
52476099260642010974402310402432010246436343
6470458815942637631277040877952013862401167364096148035889
78938717391317284668831969264291919040614566407762944544768163843404825447
8169835630412860389564821076554688887433881623353523203933224964140236824903686553678310985281
932268769777961140826947651487012166425291865625679869049344168145367042359934976212926464112066562621441801152661463
10613106625780112907507911160122281653895206864934955520252918656256063895239475211209691136013485342720106463232049807360104857641426511213649
11116490258898219269766915343244283965174045520179346425712960755142845107202225684175052846856508948487046091571207416938496052048691202191523844194304952809757913927
1222133149190661615591532427114512647440596823785645434427847283202152157108555520724937131302912178054734004224321301775646724227654942720395570053120249833717769563013121677721621914624432020321
1342052983462257059115092375791440372145379843104977312112223036782789504590647562014681602238243392897740862827884712919041322692309745664208846154170368244264507801602056964276224118103212032414397235267108864504036361936467383
1479900668578288412123549670738863953043222586522160330416271375707129290982415711225149590530566615252694564433922089027166704558085026230777033523292588461682196481299487181504512136788124368896104718181335045511483228161785095782426843545611592836324538749809
Technical details
Node id
n-8a73b9e89e682da5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-7d4eea59e559ed84

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:           19,            4;
  2:          361,          152,           16;
  3:         6859,         4332,          912,           64;
  4:       130321,       109744,        34656,         4864,         256;
  5:      2476099,      2606420,      1097440,       231040,       24320,        1024;
  6:     47045881,     59426376,     31277040,      8779520,     1386240,      116736,       4096;
  7:    893871739,   1317284668,    831969264,    291919040,    61456640,     7762944,     544768,     16384;
  8:  16983563041,  28603895648,  21076554688,   8874338816,  2335352320,   393322496,   41402368,   2490368,    65536;
  9: 322687697779, 611408269476, 514870121664, 252918656256, 79869049344, 16814536704, 2359934976, 212926464, 11206656, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (19+4*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[(19 + 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (19 + 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

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.