Real-rooted atlas

A038324: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.

A038324 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 A038324
\(P_{n}(x) = \left(11 + 10\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1111021
2121220100441
313313630330010009261
41464153240726004400010000194481
5161051732050133100012100005500001000004084101
6177156196630602196150026620000181500006600000100000085766121
71948717112400927033820710051243500046585000025410000077000000100000001801088541
8214358881155897368049603708009018856000102487000007453600000338800000088000000010000000037822859361
9235794769119292299290701538156001488111240002029242600001844766000001118040000004356000000099000000001000000000794280046581
1025937424601235794769100964614964500233846052000037202781000004058485200000307461000000015972000000005445000000001100000000001000000000016679880978201
1128531167061128531167061101296871230050035369215365000643076643000008184611820000074405562000000483153000000002196150000000066550000000001210000000000100000000000350277500542221
123138428376721342374004733201711870023666005187484920200001061076460950000154338394320000016369223640000001275523920000000724729500000000292820000000000798600000000001320000000000010000000000007355827511386641
133452271214393140799568897373022254310307658007418103435886000168593259906500002758798798470000033439985436000000303999867600000002072726370000000010468315000000000380666000000000094380000000000014300000000000010000000000000154472377739119461
14379749833583241483317970015034028559698228161100103853448102404000259633620256010000472061127738200000643719719643000000668799708720000000531999768300000000322424102000000000146556410000000000484484000000000001101100000000000015400000000000001000000000000003243919932521508681

Generated b-files: A009965 (row sums), A038313 (reversed triangle), A038324 ?

Computed in your browser ?

Technical details
Node id
n-e1d1088f7d1ff894 · revision 2 · JSON
Definition source
oeis_catalog:A038324
Fingerprint
f6-c4264000c5d1dd22

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

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,          10;
  2:        121,         220,         100;
  3:       1331,        3630,        3300,         1000;
  4:      14641,       53240,       72600,        44000,        10000;
  5:     161051,      732050,     1331000,      1210000,       550000,       100000;
  6:    1771561,     9663060,    21961500,     26620000,     18150000,      6600000,      1000000;
  7:   19487171,   124009270,   338207100,    512435000,    465850000,    254100000,     77000000,    10000000;
  8:  214358881,  1558973680,  4960370800,   9018856000,  10248700000,   7453600000,   3388000000,   880000000,  100000000;
  9: 2357947691, 19292299290, 70153815600, 148811124000, 202924260000, 184476600000, 111804000000, 43560000000, 9900000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (11+10*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 + 10*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 + 10*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.

A038313: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         10,         11;
  2:        100,        220,         121;
  3:       1000,       3300,        3630,         1331;
  4:      10000,      44000,       72600,        53240,        14641;
  5:     100000,     550000,     1210000,      1331000,       732050,       161051;
  6:    1000000,    6600000,    18150000,     26620000,     21961500,      9663060,      1771561;
  7:   10000000,   77000000,   254100000,    465850000,    512435000,    338207100,    124009270,    19487171;
  8:  100000000,  880000000,  3388000000,   7453600000,  10248700000,   9018856000,   4960370800,  1558973680,   214358881;
  9: 1000000000, 9900000000, 43560000000, 111804000000, 184476600000, 202924260000, 148811124000, 70153815600, 19292299290, 2357947691;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+10*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 + 10*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 + 10*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.

A038324: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,          10;
  2:        121,         220,         100;
  3:       1331,        3630,        3300,         1000;
  4:      14641,       53240,       72600,        44000,        10000;
  5:     161051,      732050,     1331000,      1210000,       550000,       100000;
  6:    1771561,     9663060,    21961500,     26620000,     18150000,      6600000,      1000000;
  7:   19487171,   124009270,   338207100,    512435000,    465850000,    254100000,     77000000,    10000000;
  8:  214358881,  1558973680,  4960370800,   9018856000,  10248700000,   7453600000,   3388000000,   880000000,  100000000;
  9: 2357947691, 19292299290, 70153815600, 148811124000, 202924260000, 184476600000, 111804000000, 43560000000, 9900000000, 1000000000;
%F Formula
The row polynomials satisfy P(n,x) = (11+10*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 + 10*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 + 10*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.

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-0ddfaa48a1d96c8a
  • 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-0ddfaa48a1d96c8a, e-0f31cc897fc45a50, e-96e2fb6c9c1a89f9
  • 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-0ddfaa48a1d96c8a, e-0f31cc897fc45a50, e-96e2fb6c9c1a89f9
  • 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-0ddfaa48a1d96c8a
  • 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-0f31cc897fc45a50, e-96e2fb6c9c1a89f9
  • 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-cdb8a48ab65be945
  • 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-cdb8a48ab65be945

Proof chains

4 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 7 proof steps

Same fingerprint (2)

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