A135838 → x ↦ x+1

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No OEIS matches.

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,     2;
  2:     8,     8,      2;
  3:    32,    48,     24,      4;
  4:    64,   128,     96,     32,      4;
  5:   256,   640,    640,    320,     80,     8;
  6:   512,  1536,   1920,   1280,    480,    96,     8;
  7:  2048,  7168,  10752,   8960,   4480,  1344,   224,   16;
  8:  4096, 16384,  28672,  28672,  17920,  7168,  1792,  256,  16;
  9: 16384, 73728, 147456, 172032, 129024, 64512, 21504, 4608, 576, 32;

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

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(8 + 8*x + 2*x^2)*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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 8*x + 2*x^2)*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, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f48c153b749c881a
• 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-7cade0dc9a919c2f, e-f48c153b749c881a
• 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-7cade0dc9a919c2f, e-f48c153b749c881a
• 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-7cade0dc9a919c2f

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-f48c153b749c881a

  Geometric basis map (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • preserver_image reversal, shift 0 e-07f7d31230240b98

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-51cb02553cf4a8ac root: theorem check

          Parent-row provers of polytool polytool ?

    • ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)
• preserver_image affine, shift 0 e-7cade0dc9a919c2f

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ preserver_image reversal, shift 0 e-07f7d31230240b98 (expanded above)
    • ↵ parent_provers periodic_product e-51cb02553cf4a8ac (expanded above)

Related sequences ?

Same fingerprint (67)

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

• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
• A007318 → ×(1+2n+n²) → x ↦ x+1
• basis-map coefficients of A113214, shifted → basis A013611
• A085881 → basis A013611
• A196347 → basis A007318
• A028326 → x ↦ x/(x+1)
• A038243 → basis A013610
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-…
• A156992 → basis A007318
• A028326 → basis A007318
• A038330 → basis A013609
• A120909 → basis A013610
• A178820 → x ↦ x/(x+1)
• basis-map coefficients of A112227 → basis A013609
• A135838 → x ↦ x/(x+1)
• A344565 → basis A007318
• A038338 → basis A007318
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
• A178820 → basis A007318
• A136572 → basis A013609
• A130123 → basis A013609
• basis-map coefficients of A026729 → basis A013609
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maxi…
• A027466 → basis A013611
• A178822 → basis A013611
• A013609 → D
• A344565 → basis A013611
• A159854 → x ↦ x/(x+1)
• A178822 → x ↦ x+1
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
• A135838 → basis A013611
• basis-map coefficients of A021012 → basis A013609
• A028326 → basis A013611
• basis-map coefficients of A113214, shifted → x ↦ x+1
• A038207 → D
• A196347 → basis A013611
• A178820 → basis A013611
• A356546 → basis A007318
• A178821 → basis A007318
• A007318 → D → x ↦ x/(x+1)
• A120909 → D
• A344565 → x ↦ x/(x+1)
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
• A178821 → basis A013611
• A007318 → ×(1+2n+n²) → x ↦ x/(x+1)
• A156992 → basis A013611
• A120910 → D
• basis-map coefficients of A026729, shifted → basis A013609
• A038208 → x ↦ x/(x+1)
• A356546 → basis A013611
• A085880 → x ↦ x+1
• A038208 → basis A007318
• A178821 → x ↦ x/(x+1)
• A038221 → basis A013611
• A038338 → basis A013611
• A085880 → x ↦ x/(x+1)
• A178822 → x ↦ x/(x+1)
• A085881 → basis A007318
• basis-map coefficients of A030528, shifted → basis A013609
• A013609 → basis A013610
• A038325 → basis A013611
• A038325 → basis A007318
• A038208 → basis A013611
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
• A085880 → basis A013611