Real-rooted atlas

A034870: Even-numbered rows of Pascal's triangle.

A034870 palindromic γ-positive nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · Hadamard kernel what this means

Recurrence from OEIS A034870
\(P_{n}(x) = \left(1 + 2\,x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11214
21464116
3161520156164
418285670562881256
511045120210252210120451011024
611266220495792924792495220661214096
71149136410012002300334323003200210013649114116384
811612056018204368800811440128701144080084368182056012016165536
9118153816306085681856431824437584862043758318241856485683060816153181262144
1012019011404845155043876077520125970167960184756167960125970775203876015504484511401902011048576
111222311540731526334746131705443197704974206466467054326466464974203197701705447461326334731515402312214194304
121242762024106264250413459634610473547113075041961256249614427041562496144196125613075047354713461041345964250410626202427624116777216
13126325260014950657802302306578001562275312455053117357726160965770010400600965770077261605311735312455015622756578002302306578014950260032526167108864
141283783276204759828037674011840403108105690690013123110214741803042175537442160401166003744216030421755214741801312311069069003108105118404037674098280204753276378281268435456

Generated b-files: A000302 (row sums), A034870, A034870 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-08f49539099949d5 · revision 3 · JSON
Definition source
oeis_catalog:A034870
Fingerprint
f6-2cd9f46ca2d53115

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,  2,   1;
  2: 1,  4,   6,   4,    1;
  3: 1,  6,  15,  20,   15,    6,     1;
  4: 1,  8,  28,  56,   70,   56,    28,     8,     1;
  5: 1, 10,  45, 120,  210,  252,   210,   120,    45,    10,     1;
  6: 1, 12,  66, 220,  495,  792,   924,   792,   495,   220,    66,    12,     1;
  7: 1, 14,  91, 364, 1001, 2002,  3003,  3432,  3003,  2002,  1001,   364,    91,   14,    1;
  8: 1, 16, 120, 560, 1820, 4368,  8008, 11440, 12870, 11440,  8008,  4368,  1820,  560,  120,  16,   1;
  9: 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x+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[(1 + 2*x + x^2)*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] = (1 + 2*x + x^2)*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.

A034870: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  2,   1;
  2: 1,  4,   6,   4,    1;
  3: 1,  6,  15,  20,   15,    6,     1;
  4: 1,  8,  28,  56,   70,   56,    28,     8,     1;
  5: 1, 10,  45, 120,  210,  252,   210,   120,    45,    10,     1;
  6: 1, 12,  66, 220,  495,  792,   924,   792,   495,   220,    66,    12,     1;
  7: 1, 14,  91, 364, 1001, 2002,  3003,  3432,  3003,  2002,  1001,   364,    91,   14,    1;
  8: 1, 16, 120, 560, 1820, 4368,  8008, 11440, 12870, 11440,  8008,  4368,  1820,  560,  120,  16,   1;
  9: 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+2*x+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[(1 + 2*x + x^2)*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] = (1 + 2*x + x^2)*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.

A034870: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  2,   1;
  2: 1,  4,   6,   4,    1;
  3: 1,  6,  15,  20,   15,    6,     1;
  4: 1,  8,  28,  56,   70,   56,    28,     8,     1;
  5: 1, 10,  45, 120,  210,  252,   210,   120,    45,    10,     1;
  6: 1, 12,  66, 220,  495,  792,   924,   792,   495,   220,    66,    12,     1;
  7: 1, 14,  91, 364, 1001, 2002,  3003,  3432,  3003,  2002,  1001,   364,    91,   14,    1;
  8: 1, 16, 120, 560, 1820, 4368,  8008, 11440, 12870, 11440,  8008,  4368,  1820,  560,  120,  16,   1;
  9: 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 48620, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*x+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[(1 + 2*x + x^2)*P[n - 1, x]];
Table[Reverse[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] = (1 + 2*x + x^2)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); 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-0029b53144d984cf, e-a3e5463f21dc8d9d, e-fc612a08cfe0bbb6
  • 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-0029b53144d984cf, e-5b79b5b4d4f1c577, e-a3e5463f21dc8d9d, e-fc612a08cfe0bbb6
  • 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-0029b53144d984cf, e-5b79b5b4d4f1c577, e-a3e5463f21dc8d9d, e-fc612a08cfe0bbb6
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-5b79b5b4d4f1c577

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 15 proof steps

Same fingerprint (3)

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