Real-rooted atlas

A038335 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(21 + 60\,x + 39\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213960
2441163815213600
39261515979582359319216000
4194481144471640245664982796231344112960000
540841013792379514085981026159679024291130590224199777600000
6857661219556796344437084015109870651801530341221511368249074351874376146656000000
71801088541234141510331304502700414037746453657498671985358355663069395172553328671372310066792799360000000
837822859361561939624792365260756114813566828084264314944223384704679171318858443449447960828230548091220725352009260481167961600000000
979428004658113275823635711986204041509964273550846543161190489164394166221090844816059427373152215321642178679462035804101152975023090920872836115875910077696000000000
10166798809782013097692181665902588785608963645128206525396294804166712075379581092858154822744948143709049130438610152507562342506280106210623774245445438329558433393908140406085191601604661760000000000
11350277500542221715566893964822966445497296733555370196342081801235137501498487526173035750389606756804986639358069826263782880731172527973767081782180306168992655062706399905699545188043380567925983131747583732247243936279705600000000000
1273558275113866411639298702537594281674426531877685586103654975782904345804331297202357074449512870140258432449792827885303893270307884444388851095409877856851522773592886465369542526733759207876178023693465951558673870680003911005263054628123815576555764251212176782336000000000000
13154472377739119461372940454827302698741556222109328014998282978083887328864034131382681804831258301543919353631900734917931087526851837542007491620196927248411494424844281314343817160100917472902449579065937549148521561054015918393222246109205338522184069567223380165239972364058033482880748567480579719130606940160000000000000
1432439199325215086818434191824555922570610181274416785363674517563232423897698729636386265084506203899406411434698885308757340652383996661180502966877531638482709444332827658434481378440284704084494495603170664035249077127889931815847374701700019018350811070212317517403881758756496884290275937516530851141966940078839290437386188323491941317426090417836416409600000000000000
Technical details
Node id
n-d673a74435d201fa · 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:           21,             39;
  2:          441,           1638,           1521;
  3:         9261,          51597,          95823,           59319;
  4:       194481,        1444716,        4024566,         4982796,          2313441;
  5:      4084101,       37923795,      140859810,       261596790,        242911305,         90224199;
  6:     85766121,      955679634,     4437084015,     10987065180,      15303412215,      11368249074,       3518743761;
  7:   1801088541,    23414151033,   130450270041,    403774645365,     749867198535,     835566306939,     517255332867,     137231006679;
  8:  37822859361,   561939624792,  3652607561148,  13566828084264,   31494422338470,   46791713188584,   43449447960828,   23054809122072,    5352009260481;
  9: 794280046581, 13275823635711, 98620404150996, 427355084654316, 1190489164394166, 2210908448160594, 2737315221532164, 2178679462035804, 1011529750230909, 208728361158759;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (21+60*x+39*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[((21 + 60*x + 39*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] = ((21 + 60*x + 39*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.