Real-rooted atlas

A038324 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(20 + 21\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1202141
24008404411681
380002520026460926168921
416000067200010584007408801944812825761
53200000168000003528000037044000194481004084101115856201
664000000403200000105840000014817600001166886000490092120857661214750104241
71280000000940800000029635200000518616000005445468000034306448400120072569401801088541194754273881
825600000000215040000000790272000000165957120000021781872000001829677248000960580555200288174166560378228593617984925229121
951200000000048384000000002032128000000049787136000000784147392000008233547616000057634833312000259356749904006808114684980794280046581327381934393961
1010240000000000107520000000000508032000000000142248960000000026138246400000003293419046400000288174166560000017290449993600006808114684980001588560093162001667988097820113422659310152401
1120480000000000023654400000000001241856000000000039118464000000000821487744000000001207586983680000001267966332864000009509747496480000049926174356520000174741610247820003669573815204220350277500542221550329031716248441
12409600000000000051609600000000000298045440000000000104315904000000000024644632320000000004140298229760000000507186533145600000045646787983104000002995570461391200000139793288198256000044034885782450640084066600130133040735582751138664122563490300366186081
13819200000000000001118208000000000000704471040000000000027122135040000000000711956044800000000001345596924672000000001883835694540800000001978027479267840000001557696639923424000009086563732886640000038163567678123888000109286580169172952001912515152960526660154472377739119461925103102315013629321
141638400000000000000240844800000000000001643765760000000000006903816192000000000001993476925440000000000418630154342400000000065934249308928000000007912109917071360000000726925098630931200000050884756904165184000002671449737468672160000102000808157894755200026775212141447373240043252265766953449080324391993252150868137929227194915558802161
Technical details
Node id
n-31f62c5b893224b0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-640a8dc8403e6c62

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:           20,            21;
  2:          400,           840,            441;
  3:         8000,         25200,          26460,           9261;
  4:       160000,        672000,        1058400,         740880,         194481;
  5:      3200000,      16800000,       35280000,       37044000,       19448100,        4084101;
  6:     64000000,     403200000,     1058400000,     1481760000,     1166886000,      490092120,       85766121;
  7:   1280000000,    9408000000,    29635200000,    51861600000,    54454680000,    34306448400,    12007256940,     1801088541;
  8:  25600000000,  215040000000,   790272000000,  1659571200000,  2178187200000,  1829677248000,   960580555200,   288174166560,   37822859361;
  9: 512000000000, 4838400000000, 20321280000000, 49787136000000, 78414739200000, 82335476160000, 57634833312000, 25935674990400, 6808114684980, 794280046581;
%F Formula
The row polynomials satisfy P(n,x) = (20+21*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[(20 + 21*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] = (20 + 21*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. [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.