A130123 → reverse

OEIS matches ?

• A034008: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A011782: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A166444: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A118655: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A131577: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A050732: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A220051: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A000079: 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:   2;
  2:   4;
  3:   8;
  4:  16;
  5:  32;
  6:  64;
  7: 128;
  8: 256;
  9: 512;

The row polynomials satisfy P(n,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[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] = 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-3d2f34e0586880e9, e-6e9aeea0ae5cf731, e-9754e3fde53fc70a
• 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-3d2f34e0586880e9, e-6e9aeea0ae5cf731, e-9754e3fde53fc70a
• 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-3d2f34e0586880e9, e-6e9aeea0ae5cf731, e-9754e3fde53fc70a

Proof chains

3 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

• gamma_polynomial scale 1, shift 0 e-6e9aeea0ae5cf731

  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

  • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • theorem_check liu_wang e-4f4e9585c5795aa9 root: theorem check

      Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

    • parent_provers periodic_product e-0d998323de038265 root: theorem check

      Parent-row provers of polytool polytool ?

    • basis_map A007318 e-1de26a40255316f7

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-7963c2f0e7b86ba8 root: theorem check

          Parent-row provers of polytool polytool ?

        • preserver_image mobius, shift 0 e-37a63c273567eea5

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)
    • preserver_image reversal, shift 0 e-cf1dcb38aee4f2af

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ theorem_check liu_wang e-4f4e9585c5795aa9 (expanded above)
        • ↵ parent_provers periodic_product e-0d998323de038265 (expanded above)
        • ↵ basis_map A007318 e-1de26a40255316f7 (expanded above)
        • basis_map A013609 e-e548dab430fb29bc

          Geometric basis map (elementary) polytool ?

          • premise (sequence): binomial
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • theorem_check liu_wang e-317eb4ccca60c6ac root: theorem check

              Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

            • preserver_image reversal, shift 0 e-033cf842aad3bdf7

              Affine, reversal and Möbius substitutions (elementary) polytool ?

              • premise (sequence): binomial
                uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                • ↵ theorem_check liu_wang e-317eb4ccca60c6ac (expanded above)
                • basis_map A007318 e-095baa2a9dfe188f

                  Geometric basis map (elementary) polytool ?

                  • 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 (-inf, 0]
                    • veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d

                      Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] 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)
                    • parent_provers periodic_product e-b7eca039219a7f53 root: theorem check

                      Parent-row provers of polytool polytool ?

                    • preserver_image mobius, shift 0 e-7e44b66e04c497d6

                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                      • 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 (-inf, 0]
                        • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                        • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                • parent_provers periodic_product e-13983862a411779c root: theorem check

                  Parent-row provers of polytool polytool ?

                • preserver_image reversal, shift 0 e-989c95e5c893fce9

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • premise (sequence): Riordan array (1-x,x/(1-x)).
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • parent_provers periodic_product e-b62074ba45057e82 root: theorem check

                      Parent-row provers of polytool polytool ?

                • gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea

                  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

                  • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
                    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                    • preserver_image reversal, shift 0 e-0e058c428dfdfeb6

                      Affine, reversal and Möbius substitutions (elementary) polytool ?

                      • premise (sequence): Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
                        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
                        • parent_provers periodic_product e-d82c9b48e4b31a6b root: theorem check

                          Parent-row provers of polytool polytool ?

                    • ↵ parent_provers periodic_product e-d82c9b48e4b31a6b (expanded above)
                • preserver_image affine, shift 0 e-3a52de7742ed9314

                  Affine, reversal and Möbius substitutions (elementary) polytool ?

                  • 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 (-inf, 0]
                    • ↵ veronese_section r=2, j=0, shift 0 e-292206ad4ae86d2d (expanded above)
                    • ↵ parent_provers periodic_product e-b7eca039219a7f53 (expanded above)
                    • ↵ preserver_image mobius, shift 0 e-7e44b66e04c497d6 (expanded above)
            • ↵ basis_map A007318 e-095baa2a9dfe188f (expanded above)
            • ↵ parent_provers periodic_product e-13983862a411779c (expanded above)
            • ↵ preserver_image reversal, shift 0 e-989c95e5c893fce9 (expanded above)
            • ↵ gamma_polynomial scale 1, shift 0 e-c76adc2a99cd3dea (expanded above)
            • ↵ preserver_image affine, shift 0 e-3a52de7742ed9314 (expanded above)
        • preserver_image affine, shift 0 e-07e3ff75f72fdf91

          Affine, reversal and Möbius substitutions (elementary) polytool ?

          • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
            uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
            • ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)
            • ↵ preserver_image mobius, shift 0 e-37a63c273567eea5 (expanded above)
    • ↵ basis_map A013609 e-e548dab430fb29bc (expanded above)
    • ↵ preserver_image affine, shift 0 e-07e3ff75f72fdf91 (expanded above)
• gamma_polynomial scale 1, shift 0 e-9754e3fde53fc70a

  Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

  • premise (sequence): 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.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • parent_provers periodic_product e-4755dd2fd7289161 root: theorem check

      Parent-row provers of polytool polytool ?

• preserver_image reversal, shift 0 e-3d2f34e0586880e9

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed …
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • ↵ parent_provers periodic_product e-7963c2f0e7b86ba8 (expanded above)
    • ↵ preserver_image mobius, shift 0 e-37a63c273567eea5 (expanded above)

Related sequences ?

Same fingerprint (75)

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

• basis-map coefficients of A021012 → reverse
• A085881 → γ
• A007318 → D → γ
• basis-map coefficients of A026729, shifted → θ+1
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
• A010054 → reverse → D
• basis-map coefficients of A112227 → θ+1
• A023531 → 2D+1+2n → Veronese 2 (j=1)
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
• basis-map coefficients of A030528, shifted → reverse
• basis-map coefficients of A026729 → D
• A130123 → (θ+1)²
• A131431: 3n + 1 preceded by n zeros.
• basis-map coefficients of A030528, shifted → D
• A010054 → reverse → θ+1
• basis-map coefficients of A026729, shifted → reverse
• A023531 → (θ+1)² → reverse
• A010054 → reverse → θ+1
• basis-map coefficients of A026729 → (θ+1)²
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
• A196347 → γ
• basis-map coefficients of A297191, shifted → (θ+1)²
• A136572 → θ+1
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• basis-map coefficients of A112227 → D
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
• A178820 → γ
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
• A156992 → γ
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A0082…
• A130123 → D
• basis-map coefficients of A297191, shifted → D
• A010054 → reverse → θ+1
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
• basis-map coefficients of A scaled Hermite triangle.
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros …
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downwar…
• A023531 → (θ+1)² → (θ+1)²
• basis-map coefficients of A030528, shifted → (θ+1)²
• A010054 → reverse → (θ+1)²
• A136572 → D
• A136572 → (θ+1)²
• A023531 → θ+1
• A010054 → reverse → D
• A133080 → Veronese 2
• basis-map coefficients of A112227 → (θ+1)²
• A023531 → 2D+1+2n → Veronese 2
• A023531 → (θ+1)² → D
• A010054 → reverse → reverse
• basis-map coefficients of A026729, shifted → D
• A130123 → θ+1
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
• A010054 → reverse → (θ+1)²
• A038338 → γ
• A356546 → γ
• A038221 → γ
• basis-map coefficients of A021012 → θ+1
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
• A010054 → reverse → reverse
• A132440: Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the clas…
• A178822 → γ
• A135838 → γ
• basis-map coefficients of A026729, shifted → (θ+1)²
• A028326 → γ
• A344565 → γ
• A010054 → reverse → D
• basis-map coefficients of A297191, shifted → θ+1
• basis-map coefficients of A021012 → (θ+1)²
• basis-map coefficients of A030528, shifted → θ+1
• A038325 → γ
• A023531 → (θ+1)² → θ+1
• basis-map coefficients of A021012 → D
• A178821 → γ
• A023531 → D → (θ+1)²

References

1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
3. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
4. [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