Real-rooted atlas

A038301 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(40 + 44\,x + 20\,n + 22\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 880 + 968\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
08809681848
1264005808031944116424
2704000232320025555209370246519744
317600000774400001277760009370240025768160342286560
4422400000232320000051110400005622144000309217920068027942417251242624
59856000000650496000001788864000002623667200002164525440009523911936017460505216845310888576
62252800000001734656000000572436480000010494668800000115441356800007619129548800279368083456043900698828840574922651648
750688000000004460544000000017173094400000037780807680000051948610560000045714777292800025143127511040079021257891840108654229601281917165095290368
811264000000000011151360000000004906598400000000125936025600000002077944422400000022857388646400000167620850073600007902125789184000217308459202560026559922791424089467704446883840
924780800000000002725888000000000013493145600000000039579893760000000076191295488000000010057251004416000009219146754048000005794892245401600002390393051228160005843183014113280064275013155246084133407945446033408
1054067200000000000654213120000000000359817216000000000011873968128000000000261227298816000000004022900401766400000044251904419430400000347693534724096000001912314440982528000070118196169359360001542600315725905920154260031572590592189385236773163712512
111171456000000000000154632192000000000009355247616000000000034302574592000000000084898872115200000000014942201492275200000001917582524841984000000180800638056529920000012430043866386432000006076910334677811200002005380410443677696004010760820887355392036765307524800757768617028273178948919296
122523136000000000000036080844800000000000023813357568000000000009604720885760000000000264129824358400000000005229770522296320000000076703300993679360000000843736310930472960000006960824565176401920000042538372342744678400000187168838308076584960005615065149242297548800102942861069442121728087105497827989487616389754817279170920349696
1354067200000000000000083263488000000000000005953339392000000000000026194693324800000000000079238947307520000000000017432568407654400000000002876373787262976000000000361601276113059840000000034804122825882009600000002552302340564680704000000140376628731057438720000056150651492422975488000015441429160416318259200026131649348396846284800205320102023118077952017538966777562691415736320
1411534336000000000000000190316544000000000000000146543738880000000000000069852515532800000000000002305133012582400000000000055784218904494080000000000102271067991572480000000000144640510445223936000000000159104561489746329600000000136122791496782970880000000898410423878767607808000004492052119393838039040000016470857771110739476480000418106389574349540556800065702432647397784944640048181783941425042292736785745711634808575424987136
Technical details
Node id
n-5bdb8d0739751ee3 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c4264000c5d1dd22

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:              880,               968;
  1:            26400,             58080,              31944;
  2:           704000,           2323200,            2555520,             937024;
  3:         17600000,          77440000,          127776000,           93702400,           25768160;
  4:        422400000,        2323200000,         5111040000,         5622144000,         3092179200,           680279424;
  5:       9856000000,       65049600000,       178886400000,       262366720000,       216452544000,         95239119360,        17460505216;
  6:     225280000000,     1734656000000,      5724364800000,     10494668800000,     11544135680000,       7619129548800,      2793680834560,       439006988288;
  7:    5068800000000,    44605440000000,    171730944000000,    377808076800000,    519486105600000,     457147772928000,    251431275110400,     79021257891840,     10865422960128;
  8:  112640000000000,  1115136000000000,   4906598400000000,  12593602560000000,  20779444224000000,   22857388646400000,  16762085007360000,   7902125789184000,   2173084592025600,   265599227914240;
  9: 2478080000000000, 27258880000000000, 134931456000000000, 395798937600000000, 761912954880000000, 1005725100441600000, 921914675404800000, 579489224540160000, 239039305122816000, 58431830141132800, 6427501315524608;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (20*n+40+(22*n+44)*x)*P(n-1,x) for n >= 1, with P(0,x) = 880 + 968*x.
%t Mathematica
P[0, x_] := 880 + 968*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 40 + (22*n + 44)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 880 + 968*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n + 40 + (22*n + 44)*x)*v[i - 1])/(n + 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.