Real-rooted atlas

A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.

A038325 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0) · coefficients of one sign · interlacing (weak) · Hadamard kernel basis map preserves real roots what this means

Recurrence from OEIS A038325
\(P_{n}(x) = \left(11 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111122
2121242121484
3133139933993133110648
41464158564878465856414641234256
5161051805255161051016105108052551610515153632
6177156110629366265734153543122026573415106293661771561113379904
719487171136410197409230591682050985682050985409230591136410197194871712494357888
8214358881171487104860020486681200409733615005121670120040973366002048668171487104821435888154875873536
923579476912122152921984886116876198067606044297101409066297101409066198067606044848861168762122152921923579476911207269217792
102593742460125937424601011671841070453112490952120544685916621065362309994525446859166210311249095212011671841070452593742460102593742460126559922791424
1128531167061131384283767211569214188360547076425650815941528513016301318139918222821318139918222829415285130163047076425650815156921418836053138428376721285311670611584318301411328
123138428376721376611405206522071362728635866904542428786201553522046476895248563527436303228999078200902042485635274363032155352204647689569045424287862020713627286358637661140520652313842837672112855002631049216
1334522712143931448795257871103269277154722661898734956731642662468373918291066544430730529239197592409740389855965924097403898559644430730529239197246837391829106659873495673164266269277154722661844879525787110334522712143931282810057883082752
143797498335832415316497670165374345572348560749311382289394242997243801295834168242417602591668336484821140388750250472723130330142885768311211403887502504727237602591668336484823801295834168242411382289394242997243455723485607493153164976701653743797498335832416221821273427820544

Generated b-files: A009966 (row sums), A038325, A038325 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-8064e72cf80883a7 · revision 1 · JSON
Definition source
oeis_catalog:A038325
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         11,          11;
  2:        121,         242,         121;
  3:       1331,        3993,        3993,         1331;
  4:      14641,       58564,       87846,        58564,        14641;
  5:     161051,      805255,     1610510,      1610510,       805255,       161051;
  6:    1771561,    10629366,    26573415,     35431220,     26573415,     10629366,      1771561;
  7:   19487171,   136410197,   409230591,    682050985,    682050985,    409230591,    136410197,    19487171;
  8:  214358881,  1714871048,  6002048668,  12004097336,  15005121670,  12004097336,   6002048668,  1714871048,   214358881;
  9: 2357947691, 21221529219, 84886116876, 198067606044, 297101409066, 297101409066, 198067606044, 84886116876, 21221529219, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (11+11*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[(11 + 11*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] = (11 + 11*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)

Generated with polytool c48629374597.

A038325: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,          11;
  2:        121,         242,         121;
  3:       1331,        3993,        3993,         1331;
  4:      14641,       58564,       87846,        58564,        14641;
  5:     161051,      805255,     1610510,      1610510,       805255,       161051;
  6:    1771561,    10629366,    26573415,     35431220,     26573415,     10629366,      1771561;
  7:   19487171,   136410197,   409230591,    682050985,    682050985,    409230591,    136410197,    19487171;
  8:  214358881,  1714871048,  6002048668,  12004097336,  15005121670,  12004097336,   6002048668,  1714871048,   214358881;
  9: 2357947691, 21221529219, 84886116876, 198067606044, 297101409066, 297101409066, 198067606044, 84886116876, 21221529219, 2357947691;
%F Formula
The row polynomials satisfy P(n,x) = (11+11*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[(11 + 11*x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (11 + 11*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038325: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,          11;
  2:        121,         242,         121;
  3:       1331,        3993,        3993,         1331;
  4:      14641,       58564,       87846,        58564,        14641;
  5:     161051,      805255,     1610510,      1610510,       805255,       161051;
  6:    1771561,    10629366,    26573415,     35431220,     26573415,     10629366,      1771561;
  7:   19487171,   136410197,   409230591,    682050985,    682050985,    409230591,    136410197,    19487171;
  8:  214358881,  1714871048,  6002048668,  12004097336,  15005121670,  12004097336,   6002048668,  1714871048,   214358881;
  9: 2357947691, 21221529219, 84886116876, 198067606044, 297101409066, 297101409066, 198067606044, 84886116876, 21221529219, 2357947691;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+11*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[(11 + 11*x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (11 + 11*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-f7065dbe93a909bc
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-cbffb05414f7627e, e-f7065dbe93a909bc
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-cbffb05414f7627e, e-f7065dbe93a909bc
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-f7065dbe93a909bc
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-cbffb05414f7627e
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-eb1ef354593efa7c
  • Basis map preserves real-rootedness. Basis map u^k -> P_k sends roots in (-inf, 11] to roots in (-inf, 0]: an input root r <= 11 gives the root (r - a)/d <= 0; a larger root gives a positive root
    The linear map sending x^k to the k-th polynomial of the sequence turns polynomials with roots in one interval into real-rooted ones.
    by the cited steps: e-eb1ef354593efa7c

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 8 proof steps

Same fingerprint (59)

Equal up to content, sign, reversal and a small index shift.

References

  1. [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