A034870 → basis A013609

OEIS matches ?

• A001025: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

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:      4,       8,        4;
  2:     16,      64,       96,        64,        16;
  3:     64,     384,      960,      1280,       960,        384,         64;
  4:    256,    2048,     7168,     14336,     17920,      14336,       7168,       2048,         256;
  5:   1024,   10240,    46080,    122880,    215040,     258048,     215040,     122880,       46080,       10240,        1024;
  6:   4096,   49152,   270336,    901120,   2027520,    3244032,    3784704,    3244032,     2027520,      901120,      270336,      49152,       4096;
  7:  16384,  229376,  1490944,   5963776,  16400384,   32800768,   49201152,   56229888,    49201152,    32800768,    16400384,    5963776,    1490944,     229376,     16384;
  8:  65536, 1048576,  7864320,  36700160, 119275520,  286261248,  524812288,  749731840,   843448320,   749731840,   524812288,  286261248,  119275520,   36700160,   7864320,   1048576,    65536;
  9: 262144, 4718592, 40108032, 213909504, 802160640, 2246049792, 4866441216, 8342470656, 11470897152, 12745441280, 11470897152, 8342470656, 4866441216, 2246049792, 802160640, 213909504, 40108032, 4718592, 262144;

The row polynomials satisfy P(n,x) = (4+8*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + 8*x + 4*x^2)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4 + 8*x + 4*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)

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-9638204c7017296b
• 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-9638204c7017296b
• 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-9638204c7017296b

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

• basis_map A013609 e-9638204c7017296b

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Even-numbered rows of Pascal's triangle.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-0029b53144d984cf root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image reversal, shift 0 e-a3e5463f21dc8d9d

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Even-numbered rows of Pascal's triangle.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-0029b53144d984cf (expanded above)
        • basis_map A007318 e-fc612a08cfe0bbb6

          Geometric basis map (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • parent_provers periodic_product e-1991a148b9538b93 root: theorem check

              Parent-row provers of polytool polytool ?

            • preserver_image mobius, shift 0 e-12303ad0e0b9f8eb

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
        • preserver_image affine, shift 0 e-5b79b5b4d4f1c577

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-1991a148b9538b93 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-12303ad0e0b9f8eb (expanded above)
    • ↵ basis_map A007318 e-fc612a08cfe0bbb6 (expanded above)
    • ↵ preserver_image affine, shift 0 e-5b79b5b4d4f1c577 (expanded above)

Related sequences ?

Same fingerprint (3)

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

• A139548: Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
• A034870: Even-numbered rows of Pascal's triangle.
• A139548 → basis A013609