A034870 → basis A013610

OEIS matches ?

• A009969: 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,      12,        9;
  2:     16,      96,      216,       216,         81;
  3:     64,     576,     2160,      4320,       4860,        2916,         729;
  4:    256,    3072,    16128,     48384,      90720,      108864,       81648,        34992,         6561;
  5:   1024,   15360,   103680,    414720,    1088640,     1959552,     2449440,      2099520,      1180980,       393660,        59049;
  6:   4096,   73728,   608256,   3041280,   10264320,    24634368,    43110144,     55427328,     51963120,     34642080,     15588936,      4251528,       531441;
  7:  16384,  344064,  3354624,  20127744,   83026944,   249080832,   560431872,    960740352,   1260971712,   1260971712,    945728784,    515852064,    193444524,     44641044,      4782969;
  8:  65536, 1572864, 17694720, 123863040,  603832320,  2173796352,  5977939968,  12809871360,  21616657920,  28822210560,  30263321088,  24760899072,  15475561920,   7142567040,   2295825120,   459165024,    43046721;
  9: 262144, 7077888, 90243072, 721944576, 4060938240, 17055940608, 55431806976, 142538932224, 293986547712, 489977579520, 661469732352, 721603344384, 631402926336, 437125102848, 234174162240, 93669664896, 26344593252, 4649045868, 387420489;

The row polynomials satisfy P(n,x) = (4+12*x+9*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 + 12*x + 9*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 + 12*x + 9*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-cb7f9b19b94bd36e
• 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-cb7f9b19b94bd36e
• 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-cb7f9b19b94bd36e

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 A013610 e-cb7f9b19b94bd36e

  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 (1)

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

• A139548 → basis A013610