{"id":"n-841b8a0af1d94710","revision":0,"first_index":0,"fingerprint":"f6-1f0181c04d99414a","name":"theta_plus_one_squared of veronese_section_2_0 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-1"],["1","4"],["-1","-12"],["1","24","9"],["-1","-40","-45"],["1","60","135","16"],["-1","-84","-315","-112"],["1","112","630","448","25"],["-1","-144","-1134","-1344","-225"],["1","180","1890","3360","1125","36"],["-1","-220","-2970","-7392","-4125","-396"],["1","264","4455","14784","12375","2376","49"],["-1","-312","-6435","-27456","-32175","-10296","-637"],["1","364","9009","48048","75075","36036","4459","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","-1","5","-13","34","-86","212","-512","1216","-2848","6592","-15104","34304","-77312","173056"],"nonn":false,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-1"]],"recurrence":{"denominator":{"coeffs":[["-36","-12","0"],["36","10","0"],["0","24","-1"],["0","-22","-1"],["0","0","1"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["36","156","-4","4"],["-36","-154","-62","8"],["0","-24","29","1"],["0","22","39","-7"],["0","0","-1","-5"],["0","0","-1","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-72","88","-24","8"],["0","0","84","-108","24"],["0","0","44","-70","26"],["0","0","0","-12","12"],["0","0","0","-2","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171863,"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-01d022a27a535273"]},{"id":171864,"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-01d022a27a535273"]},{"id":171865,"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-01d022a27a535273"]}],"region_exact":{},"incoming_edges":[{"id":"e-01d022a27a535273","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-841b8a0af1d94710","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":"4da505f75c82f36525ceede3a17ebf1822c5a3bd60c0697a2f35322eacda58ce","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-d94eb58321d693e7","role":"sequence","name":"veronese_section_2_0 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","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-12c504f0404f8739","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-745a4a6b272e9e4d","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-745a4a6b272e9e4d","other":{"id":"n-745a4a6b272e9e4d","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6"],["21","84"]],"card_rows":[["1"],["6"],["21","84"],["56","672"]],"row_sums_preview":["1","6","105","728","4284","21672","97944","405504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-496b1966a8762df1","e-3a36bf9839123f38"],"text":"A178822 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A178822 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-22164d4c98329a74","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-202941dcb2123fea","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-202941dcb2123fea","other":{"id":"n-202941dcb2123fea","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["2","8"]],"card_rows":[["1"],["2"],["2","8"],["2","24"]],"row_sums_preview":["1","2","10","26","68","172","424","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ced86a7ce48c966e","steps":["(\u03b8+1)\u00b2"],"edges":["e-84b40fb5012c9317"],"text":"A119462 \u2192 (\u03b8+1)\u00b2"},"headline":"A119462 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-33cb46bafdab3de5","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-6ec34513293cdda0","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-6ec34513293cdda0","other":{"id":"n-6ec34513293cdda0","name":"theta_plus_one_squared of Table: T(n,k) = n!*binomial(n+1,2*k).","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["2","24"]],"card_rows":[["1"],["1","4"],["2","24"],["6","144","54"]],"row_sums_preview":["1","5","26","204","2064","25440","368640","6128640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-90f6949024558b8d","steps":["(\u03b8+1)\u00b2"],"edges":["e-4ed12709a7f73fb2"],"text":"A228955 \u2192 (\u03b8+1)\u00b2"},"headline":"A228955 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-5857a87eef49c437","kind":"fingerprint_related","node_a_id":"n-af0246c88c139e87","node_b_id":"n-841b8a0af1d94710","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-af0246c88c139e87","other":{"id":"n-af0246c88c139e87","name":"theta_plus_one_squared of veronese_section_2_0 of binomial","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","12"]],"row_sums_preview":["1","1","5","13","34","86","212","512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9cbb4c37ddc3c985","steps":["(\u03b8+1)\u00b2"],"edges":["e-1da06d7d5e63865c"],"text":"A034839 \u2192 (\u03b8+1)\u00b2"},"headline":"A034839 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-59e63cefaa8898c9","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-d745086d861ef817","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-d745086d861ef817","other":{"id":"n-d745086d861ef817","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4"],["10","40"]],"card_rows":[["1"],["4"],["10","40"],["20","240"]],"row_sums_preview":["1","4","50","260","1190","4816","17808","61440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-9985dc3bbd6a1733","e-4168e5c5dcbea788"],"text":"A178820 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A178820 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-6e18d0d89fdde7c2","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-094763b148064a77","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-094763b148064a77","other":{"id":"n-094763b148064a77","name":"theta_plus_one_squared of veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2"],["2","8"]],"card_rows":[["2"],["2"],["2","8"],["2","24"]],"row_sums_preview":["2","2","10","26","68","172","424","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-e811ab86eff6d185","e-1fe02b877a13aae0"],"text":"A028326 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A028326 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-cc87e93f4219ad94","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-1505e8f7a40e80bf","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-1505e8f7a40e80bf","other":{"id":"n-1505e8f7a40e80bf","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5"],["15","60"]],"card_rows":[["1"],["5"],["15","60"],["35","420"]],"row_sums_preview":["1","5","75","455","2380","10836","44520","168960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-8ba085c2fb60bb1b","e-8504682fd01917c7"],"text":"A178821 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A178821 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-cf40794a28811578","kind":"fingerprint_related","node_a_id":"n-841b8a0af1d94710","node_b_id":"n-2a832b7b6a769b3d","fingerprint":"f6-1f0181c04d99414a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1f0181c04d99414a"},"other_id":"n-2a832b7b6a769b3d","other":{"id":"n-2a832b7b6a769b3d","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-1f0181c04d99414a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","16"]],"card_rows":[["1"],["2"],["4","16"],["8","96"]],"row_sums_preview":["1","2","20","104","544","2752","13568","65536"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-01a436d0e1f57933","e-46a525a2e4f7215b"],"text":"A038208 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A038208 \u2192 Veronese 2 \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: -1;\n  2:  1,    4;\n  3: -1,  -12;\n  4:  1,   24,     9;\n  5: -1,  -40,   -45;\n  6:  1,   60,   135,    16;\n  7: -1,  -84,  -315,  -112;\n  8:  1,  112,   630,   448,   25;\n  9: -1, -144, -1134, -1344, -225;\n","recurrence_sentence":"The row polynomials satisfy (36*n-36-(22*n^3-24*n^2-10*n+12)*x+(n^5+n^4-n^3-n^2)*x^2)*P(n,x) = -(36*n-36-(22*n^3-24*n^2-154*n+156)*x+(n^5+n^4-39*n^3-29*n^2+62*n+4)*x^2+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^3)*P(n-1,x) - (72*x-(44*n^2+84*n+88)*x^2+(2*n^4+12*n^3+70*n^2+108*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 >= 2, with P(0,x) = 1, P(1,x) = -1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(36*n - 36 - (22*n^3 - 24*n^2 - 154*n + 156)*x + (n^5 + n^4 - 39*n^3 - 29*n^2 + 62*n + 4)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*P[n - 1, x] - (72*x - (44*n^2 + 84*n + 88)*x^2 + (2*n^4 + 12*n^3 + 70*n^2 + 108*n + 24)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*D[P[n - 1, x], x])/(36*n - 36 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (n^5 + n^4 - n^3 - n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -1; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n - 36 - (22*n^3 - 24*n^2 - 154*n + 156)*x + (n^5 + n^4 - 39*n^3 - 29*n^2 + 62*n + 4)*x^2 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^3)*v[i - 1] - (72*x - (44*n^2 + 84*n + 88)*x^2 + (2*n^4 + 12*n^3 + 70*n^2 + 108*n + 24)*x^3 - (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^4)*deriv(v[i - 1], x))/(36*n - 36 - (22*n^3 - 24*n^2 - 10*n + 12)*x + (n^5 + n^4 - n^3 - n^2)*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-74c1e94669ff21ff","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-969b818b29eb86a6","e-01d022a27a535273"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}