{"id":"n-12a7b79b8c283509","revision":0,"first_index":0,"fingerprint":"f6-4b3b88e53403151e","name":"theta_plus_one_squared of Triangle of odd-numbered terms in rows of Pascal's triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["3","4"],["4","16"],["5","40","9"],["6","80","54"],["7","140","189","16"],["8","224","504","128"],["9","336","1134","576","25"],["10","480","2268","1920","250"],["11","660","4158","5280","1375","36"],["12","880","7128","12672","5500","432"],["13","1144","11583","27456","17875","2808","49"],["14","1456","18018","54912","50050","13104","686"],["15","1820","27027","102960","125125","49140","5145","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","7","20","54","140","352","864","2080","4928","11520","26624","60928","138240","311296"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["4","4","0"],["0","-2","0"],["0","-10","1"],["0","0","2"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["4","-12","12","-4"],["4","18","26","-8"],["0","-12","-19","-1"],["0","-10","-15","7"],["0","0","3","5"],["0","0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-24","24","-8"],["0","0","-44","68","-24"],["0","0","-20","46","-26"],["0","0","0","12","-12"],["0","0","0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171935,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-48e6e483df24acb8"]},{"id":171936,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-48e6e483df24acb8"]},{"id":171937,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-48e6e483df24acb8"]}],"region_exact":{},"incoming_edges":[{"id":"e-48e6e483df24acb8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-12a7b79b8c283509","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"41edaf6cce992767b9721a804a4dae2ea3a3b123562f45e7cfa74d67f3db7060","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fd133d9d2402ed18","role":"sequence","name":"Triangle of odd-numbered terms in rows of Pascal's triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-2a7b1f5de0bbae4a","kind":"fingerprint_related","node_a_id":"n-1e6dbfaf52669ddc","node_b_id":"n-12a7b79b8c283509","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-1e6dbfaf52669ddc","other":{"id":"n-1e6dbfaf52669ddc","name":"theta_plus_one_squared of veronese_section_2_1 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],["-1"],["2"]],"card_rows":[["-1"],["2"],["-3","-4"],["4","16"]],"row_sums_preview":["0","-1","2","-7","20","-54","140","-352"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-a70814c31408551f","e-948cd88b43ef175f"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-392778eb845dc32c","kind":"fingerprint_related","node_a_id":"n-12a7b79b8c283509","node_b_id":"n-d36727987a251097","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-d36727987a251097","other":{"id":"n-d36727987a251097","name":"theta_plus_one of veronese_section_2_1 of Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["6","8"],["8","32"]],"row_sums_preview":["0","2","4","14","40","108","280","704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-801895951aa20f29","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-274dee66a9035b75","e-96208235d35b2d6b"],"text":"A103406 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A103406 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-6d1e7520de64d794","kind":"fingerprint_related","node_a_id":"n-12a7b79b8c283509","node_b_id":"n-267a0fd66a751b9e","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-267a0fd66a751b9e","other":{"id":"n-267a0fd66a751b9e","name":"theta_plus_one_squared of veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["42"]],"card_rows":[["6"],["42"],["168","224"],["504","2016"]],"row_sums_preview":["0","6","42","392","2520","13608","64680","278784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-d44b85edec7c5853","e-36a23e3817afbf01"],"text":"A178822 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A178822 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-6e36466f8de9e452","kind":"fingerprint_related","node_a_id":"n-859575996a9c391a","node_b_id":"n-12a7b79b8c283509","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-859575996a9c391a","other":{"id":"n-859575996a9c391a","name":"theta_plus_one_squared of veronese_section_2_1 of binomial","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["3","4"],["4","16"]],"row_sums_preview":["0","1","2","7","20","54","140","352"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-3a4f64258abf2a3d","e-10b7db033d9f2b41"],"text":"A007318 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-7ed9427c5d2ce1c1","kind":"fingerprint_related","node_a_id":"n-cc6e33e70de51559","node_b_id":"n-12a7b79b8c283509","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-cc6e33e70de51559","other":{"id":"n-cc6e33e70de51559","name":"theta_plus_one_squared of veronese_section_2_1 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["6","8"],["8","32"]],"row_sums_preview":["0","2","4","14","40","108","280","704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-53a911614b54e844","e-b640f15271b8fc55"],"text":"A028326 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A028326 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-9290c07f748d2522","kind":"fingerprint_related","node_a_id":"n-12a7b79b8c283509","node_b_id":"n-b62e3fe53a56a2b6","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-b62e3fe53a56a2b6","other":{"id":"n-b62e3fe53a56a2b6","name":"theta_plus_one_squared of veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8"]],"card_rows":[["2"],["8"],["24","32"],["64","256"]],"row_sums_preview":["0","2","8","56","320","1728","8960","45056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-0d347b78871c0570","e-bf3bf09ada96cda7"],"text":"A038208 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A038208 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-a62f341716f2b5ee","kind":"fingerprint_related","node_a_id":"n-12a7b79b8c283509","node_b_id":"n-6905027b74f9fc28","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-6905027b74f9fc28","other":{"id":"n-6905027b74f9fc28","name":"theta_plus_one_squared of veronese_section_2_1 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["5"],["30"]],"card_rows":[["5"],["30"],["105","140"],["280","1120"]],"row_sums_preview":["0","5","30","245","1400","6804","29400","116160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-1471988cf5f0d7aa","e-fa6a217fa0bc74c1"],"text":"A178821 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A178821 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-d95062115723ffae","kind":"fingerprint_related","node_a_id":"n-12a7b79b8c283509","node_b_id":"n-8d0532fccf53f57d","fingerprint":"f6-4b3b88e53403151e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4b3b88e53403151e"},"other_id":"n-8d0532fccf53f57d","other":{"id":"n-8d0532fccf53f57d","name":"theta_plus_one_squared of veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-4b3b88e53403151e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["20"]],"card_rows":[["4"],["20"],["60","80"],["140","560"]],"row_sums_preview":["0","4","20","140","700","3024","11760","42240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-4dba393820f57c76","e-b6802e9d828ff1a5"],"text":"A178820 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A178820 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  2;\n  2:  3,   4;\n  3:  4,  16;\n  4:  5,  40,    9;\n  5:  6,  80,   54;\n  6:  7, 140,  189,   16;\n  7:  8, 224,  504,  128;\n  8:  9, 336, 1134,  576,  25;\n  9: 10, 480, 2268, 1920, 250;\n","recurrence_sentence":"The row polynomials satisfy (4*n-(10*n^3+2*n^2-4*n)*x+(n^5+2*n^4+n^3)*x^2)*P(n,x) = (4*n+4-(10*n^3+12*n^2-18*n+12)*x+(n^5+3*n^4-15*n^3-19*n^2+26*n+12)*x^2+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^3)*P(n-1,x) + (8*x-(20*n^2+44*n+24)*x^2+(2*n^4+12*n^3+46*n^2+68*n+24)*x^3-(2*n^4+12*n^3+26*n^2+24*n+8)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 4 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*P[n - 1, x] + (8*x - (20*n^2 + 44*n + 24)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(4*n - (10*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 + n^3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 4 - (10*n^3 + 12*n^2 - 18*n + 12)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*v[i - 1] + (8*x - (20*n^2 + 44*n + 24)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(4*n - (10*n^3 + 2*n^2 - 4*n)*x + (n^5 + 2*n^4 + n^3)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-fd133d9d2402ed18","steps":["(\u03b8+1)\u00b2"],"edges":["e-48e6e483df24acb8"],"text":"A034867 \u2192 (\u03b8+1)\u00b2"},"headline":"A034867 \u2192 (\u03b8+1)\u00b2","chain":null}