Real-rooted atlas

A013613 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(7 + 20\,x + 13\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
171320
249182169400
33431911354921978000
4240117836496866151628561160000
51680715606557967010765309996353712933200000
611764913109466086535150714202099233515594306482680964000000
78235431070605959648043184624895342874805382060497236513641627485171280000000
85764801856484725567150682067798824480024727071317959446622381948351391695281573072125600000000
940353607674481717501043561221711887652604831156021123257861181390700209081106883839885139103542310604499373512000000000
10282475249524596891043841311605217118876520705636348690157256100565224337253658902582728959720179868623980574231495611013785849184910240000000000
11197732674340393960607375086777065208976918650577619998355902018119957253437479370634706497175324746104616628015499528579125810235106151038723731792160394037204800000000000
121384128720130846297190831507289273461950451240738081500998273695242174394870408524711188885884835254545573448969491883254895800215522686580445834362639666150541473099108232980851224814096000000000000
139688901040723391775369692606512112622617749106290715882406564921180527547337416508916821245455516492126680272745306281764475227523908918204903141119695135236423334032026849637026009414212012574614577130287510659225381920000000000000
14678223072849176337998940742128651558641791581284014991044807584336227568929995989631309702835602568300770271773523818434287922881976204955717873568161015659460223313292371683784492237548095163075241038861615611427792968176044604079439373763856992891638400000000000000
Technical details
Node id
n-290098397986c082 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-60d4e40326205ee3

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:        7,        13;
  2:       49,       182,        169;
  3:      343,      1911,       3549,        2197;
  4:     2401,     17836,      49686,       61516,       28561;
  5:    16807,    156065,     579670,     1076530,      999635,       371293;
  6:   117649,   1310946,    6086535,    15071420,    20992335,     15594306,      4826809;
  7:   823543,  10706059,   59648043,   184624895,   342874805,    382060497,    236513641,     62748517;
  8:  5764801,  85648472,  556715068,  2067798824,  4800247270,   7131795944,   6622381948,   3513916952,   815730721;
  9: 40353607, 674481717, 5010435612, 21711887652, 60483115602, 112325786118, 139070020908, 110688383988, 51391035423, 10604499373;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (7+20*x+13*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[((7 + 20*x + 13*x^2)*P[n - 1, x])/(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] = ((7 + 20*x + 13*x^2)*v[i - 1])/(1 + 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.