A249139: Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

OEIS matches ?

• A249139: triangle, 57 terms agree, 13 full rows row lengths from the atlas (tabf) OEIS nonn in polytool catalog · view at the OEIS · generated b-file 991 terms from n = 0 ?
• A006130: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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.

Triangle begins:
  0:   1;
  1:   3,    1;
  2:   5,    2;
  3:  11,    7,   1;
  4:  21,   16,   3;
  5:  43,   41,  12,   1;
  6:  85,   94,  34,   4;
  7: 171,  219,  99,  18,  1;
  8: 341,  492, 261,  60,  5;
  9: 683, 1101, 678, 195, 25, 1;

The row polynomials satisfy P(n,x) = P(n-1,x) + (2+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + (2 + x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + (2 + x)*v[i - 2]); 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)

Triangle begins:
  0:   1;
  1:   3,    1;
  2:   5,    2;
  3:  11,    7,   1;
  4:  21,   16,   3;
  5:  43,   41,  12,   1;
  6:  85,   94,  34,   4;
  7: 171,  219,  99,  18,  1;
  8: 341,  492, 261,  60,  5;
  9: 683, 1101, 678, 195, 25, 1;

The row polynomials satisfy P(n,x) = P(n-1,x) + (2+x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + (2 + x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + (2 + x)*v[i - 2]); 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)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -2]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-7fc1b55fb0aa373a
• 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-7fc1b55fb0aa373a
• 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-7fc1b55fb0aa373a

Proof chains

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

• preserver_image affine, shift 1 e-7fc1b55fb0aa373a

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one vers…
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers paper_theorem e-0801036cdfff24d2 root: theorem check

      Parent-row provers of polytool polytool ?

    • parent_provers paper_theorem e-0d3631d0f359c9ce root: theorem check

      Parent-row provers of polytool polytool ?

    • gamma_polynomial scale 1, shift 0 e-a039cba7823b0b25

      Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

      • premise (sequence): delannoy
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang_derivative_lag e-15c233e4385971d1 root: theorem check

          Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?

        • preserver_image reversal, shift 0 e-266d85ca03a298fa

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): delannoy
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ theorem_check liu_wang_derivative_lag e-15c233e4385971d1 (expanded above)
            • basis_map A008288 e-e771e35dc9ee3fbd

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
            • basis_map A008288 e-33dba7e79fe9caa8

              Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

              • premise (sequence): a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in [0, 0]
                no recorded property covers this requirement
        • ↵ basis_map A008288 e-e771e35dc9ee3fbd (expanded above)
        • ↵ basis_map A008288 e-33dba7e79fe9caa8 (expanded above)

Used in 11 proof steps

• e-16890f4567488209 basis_map (A007318) → basis_map_A007318 of Triangular array read by rows: row n gives the coefficients of the p…
• e-35e10c7226ff172b basis_map (A008277) → basis_map_A008277 of Triangular array read by rows: row n gives the coefficients of the p…
• e-6895a0e432d3b559 basis_map (A013610) → basis_map_A013610 of Triangular array read by rows: row n gives the coefficients of the p…
• e-96071100df54473b basis_map (A013611) → basis_map_A013611 of Triangular array read by rows: row n gives the coefficients of the p…
• e-b6c71ffc5380c303 basis_map (A013609) → basis_map_A013609 of Triangular array read by rows: row n gives the coefficients of the p…
• e-14d41bb220ff7187 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangular array read by rows: row n gives the coefficients of the…
• e-42735f4be786e5b0 preserver_image (euler_polynomial, shift 0) → theta_plus_one of Triangular array read by rows: row n gives the coefficients of the poly…
• e-63cc36dbb2ec429b preserver_image (derivative_polynomial, shift 0) → derivative of Triangular array read by rows: row n gives the coefficients of the polynomi…
• e-8280e16be483f9ab preserver_image (affine, shift 0) → basis_map_A007318 of Triangular array read by rows: row n gives the coefficients of the p…
• e-0527b2f699cf5083 veronese_section (r=2, j=0, shift 0) → veronese_section_2_0 of Triangular array read by rows: row n gives the coefficients of th…
• e-d9167ed20c29eacc veronese_section (r=2, j=1, shift 0) → veronese_section_2_1 of Triangular array read by rows: row n gives the coefficients of th…

Related sequences ?

Same fingerprint (1)

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

• A011973 → basis A007318 → x ↦ x+1

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