Real-rooted atlas

A038319 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 5\,x + 5\,n\right)\,P_{n}(x) = \left(288 + 260\,x + 50\,x^{2} + 80\,n + 50\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 576 + 520\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
05765201001196
111776172807800100037856
2229376471040345600104000100001160016
3432537611468800117760005760000130000010000034730176
479691776259522560344064000235520000864000001560000010000001021798336
51442840576557842432090832896008028160000412160000012096000001820000001000000029655914496
62576980377611542724608022313697280024222105600016056320000065945600000161280000002080000000100000000851371878656
74552665333762319282339840519422607360066941091840005449973760000289013760000098918400000020736000000023400000000100000000024223939490816
87971459301376455266533376001159641169920001731408691200001673527296000001089994752000004816896000000014131200000000259200000000026000000000010000000000684117463550976
91385384650997768768605231513602503965933568000425201762304000047613739008000003681760051200000199832371200000075694080000000019430400000000031680000000000286000000000010000000000019198725008859136
10239253730204057616624615811973120526116313890816001001586373427200001275605286912000001142729736192000007363520102400000034256977920000000113541120000000002590720000000000380160000000000312000000000001000000000000535870295100215296
11410953465997557763110298492652748801080600027778252800227983736019353600032551557136384000003316573745971200000247591442841600000013675108761600000005566758912000000001640038400000000003367936000000000044928000000000003380000000000001000000000000014886917239222419456
1270256154186979737657533485239658086402177208944856924160050428001296318464000797943076067737600009114435998187520000077386720739328000000495182885683200000002393144033280000000086594027520000000002296053760000000000428646400000000000524160000000000003640000000000000100000000000000411871376951820271616
13119615606102960373761053842312804696064004315011392974356480001088604472428462080000189105004861194240000023938292282032128000002278608999546880000000165828687298560000000092846791065600000000039885733888000000000012989104128000000000031309824000000000000535808000000000000060480000000000000039000000000000000100000000000000011353755547780298571776
1420291418481080506777619138496976473659801608430738502437568512000230133940958632345600004354417889713848320000060513601555582156800000638354460854190080000005208249141821440000000033165737459712000000000165060961894400000000006381717422080000000000188932423680000000000041746432000000000000065945600000000000000691200000000000000041600000000000000010000000000000000311970237665145022119936
Technical details
Node id
n-dd032736ed5eddee · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-354a4e768d29be07

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:             576,             520,              100;
  1:           11776,           17280,             7800,             1000;
  2:          229376,          471040,           345600,           104000,            10000;
  3:         4325376,        11468800,         11776000,          5760000,          1300000,           100000;
  4:        79691776,       259522560,        344064000,        235520000,         86400000,         15600000,          1000000;
  5:      1442840576,      5578424320,       9083289600,       8028160000,       4121600000,       1209600000,        182000000,        10000000;
  6:     25769803776,    115427246080,     223136972800,     242221056000,     160563200000,      65945600000,      16128000000,      2080000000,       100000000;
  7:    455266533376,   2319282339840,    5194226073600,    6694109184000,    5449973760000,    2890137600000,     989184000000,    207360000000,     23400000000,     1000000000;
  8:   7971459301376,  45526653337600,  115964116992000,  173140869120000,  167352729600000,  108999475200000,   48168960000000,  14131200000000,   2592000000000,   260000000000,   10000000000;
  9: 138538465099776, 876860523151360, 2503965933568000, 4252017623040000, 4761373900800000, 3681760051200000, 1998323712000000, 756940800000000, 194304000000000, 31680000000000, 2860000000000, 100000000000;
%F Formula
The row polynomials satisfy (5*n+13+5*x)*P(n,x) = (80*n+288+(50*n+260)*x+50*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 576 + 520*x + 100*x^2.
%t Mathematica
P[0, x_] := 576 + 520*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((80*n + 288 + (50*n + 260)*x + 50*x^2)*P[n - 1, x])/(5*n + 13 + 5*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 576 + 520*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 288 + (50*n + 260)*x + 50*x^2)*v[i - 1])/(5*n + 13 + 5*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. [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.