A034870 → x ↦ x+1

OEIS matches ?

• A100062: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
• A001019: 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,       4,        1;
  2:     16,      32,       24,        8,        1;
  3:     64,     192,      240,      160,       60,       12,        1;
  4:    256,    1024,     1792,     1792,     1120,      448,      112,       16,        1;
  5:   1024,    5120,    11520,    15360,    13440,     8064,     3360,      960,      180,       20,        1;
  6:   4096,   24576,    67584,   112640,   126720,   101376,    59136,    25344,     7920,     1760,      264,      24,       1;
  7:  16384,  114688,   372736,   745472,  1025024,  1025024,   768768,   439296,   192192,    64064,    16016,    2912,     364,     28,     1;
  8:  65536,  524288,  1966080,  4587520,  7454720,  8945664,  8200192,  5857280,  3294720,  1464320,   512512,  139776,   29120,   4480,   480,   32,   1;
  9: 262144, 2359296, 10027008, 26738688, 50135040, 70189056, 76038144, 65175552, 44808192, 24893440, 11202048, 4073472, 1188096, 274176, 48960, 6528, 612, 36, 1;

The row polynomials satisfy P(n,x) = (4+4*x+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 + 4*x + 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 + 4*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)

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-f2e5e2864fad238b
• 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-cfe8f3b6680af0c3, e-f2e5e2864fad238b
• 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-cfe8f3b6680af0c3, e-f2e5e2864fad238b
• 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-cfe8f3b6680af0c3

Proof chains

2 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

• basis_map A007318 e-f2e5e2864fad238b

  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)
• preserver_image affine, shift 0 e-cfe8f3b6680af0c3

  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)
    • ↵ preserver_image reversal, shift 0 e-a3e5463f21dc8d9d (expanded above)
    • ↵ basis_map A007318 e-fc612a08cfe0bbb6 (expanded above)
    • ↵ preserver_image affine, shift 0 e-5b79b5b4d4f1c577 (expanded above)

Related sequences ?

Same fingerprint (5)

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

• A139548 → basis A007318
• basis-map coefficients of A297191, shifted → basis A013609
• A034870 → basis A013611
• A139548 → basis A013611
• A034870 → x ↦ x/(x+1)