{"id":"n-88948f8cc95a6e83","revision":0,"first_index":1,"fingerprint":"f6-9be755efdd908b94","name":"reversal of theta_plus_one of Triangle read by rows: 2-Stirling numbers of the second kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","2"],["3","10","4"],["4","27","38","8"],["5","56","165","130","16"],["6","100","500","855","422","32"],["7","162","1225","3640","4053","1330","64"],["8","245","2604","11900","23604","18207","4118","128"],["9","352","4998","32508","100905","142296","79005","12610","256"],["10","486","8880","77910","348642","783300","816820","334935","38342","512"],["11","650","14850","168960","1030029","3380454","5720825","4530680","1397253","116050","1024"],["12","847","23650","338580","2697816","12184095","30528168","39988300","24516404","5763087","350198","2048"],["13","1080","36179","636350","6420843","38205024","133088956","261683136","270575305","130266136","23577645","1054690","4096"],["14","1352","53508","1134133","14134120","107189082","495946880","1370008640","2156472318","1786284500","682767540","95892615","3172262","8192"],["15","1666","76895","1932840","29157128","274574300","1629831060","6029246080","13475953491","17238847626","11571166425","3541694520","388328853","9533170","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","17","77","372","1915","10481","60814","372939","2409837","16360786","116393205","865549453","6713065156","54190360453"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1","1"],["0","-1","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178949,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-86990400404fca4b"]},{"id":178950,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-86990400404fca4b"]},{"id":178951,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-86990400404fca4b"]}],"region_exact":{},"incoming_edges":[{"id":"e-86990400404fca4b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-88948f8cc95a6e83","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"cdbb40153b36a302c4b3debdd7ee96fe07e55490aad017093437cb4bc44f75b5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-62c0157c67ca5559","role":"sequence","name":"theta_plus_one of Triangle read by rows: 2-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-19191394016daba4","kind":"fingerprint_related","node_a_id":"n-d72031dd4f362d3c","node_b_id":"n-88948f8cc95a6e83","fingerprint":"f6-9be755efdd908b94","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9be755efdd908b94"},"other_id":"n-d72031dd4f362d3c","other":{"id":"n-d72031dd4f362d3c","name":"theta_plus_one of basis_map_A008277 of binomial","first_index":0,"fingerprint":"f6-9be755efdd908b94","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["4","10","3"]],"card_rows":[["1"],["2","2"],["4","10","3"],["8","38","27","4"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0990b5be9f344750","steps":["reverse","\u03b8+1"],"edges":["e-63d258a3014acdf4","e-54bf52bd4d3f711a"],"text":"A126351 \u2192 reverse \u2192 \u03b8+1"},"headline":"A126351 \u2192 reverse \u2192 \u03b8+1","chain":null}},{"id":"l-69021e56867fb93e","kind":"fingerprint_related","node_a_id":"n-88948f8cc95a6e83","node_b_id":"n-4a6fbeb2444d3f2e","fingerprint":"f6-9be755efdd908b94","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9be755efdd908b94"},"other_id":"n-4a6fbeb2444d3f2e","other":{"id":"n-4a6fbeb2444d3f2e","name":"theta_plus_one of basis_map_A008277 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-9be755efdd908b94","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["24","60","18"]],"card_rows":[["1"],["4","4"],["24","60","18"],["192","912","648","96"]],"row_sums_preview":["1","8","102","1848","44640","1378800","52824240","2452020480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A008277","\u03b8+1"],"edges":["e-c0fcd074b1f3c276","e-b34999c7d2bfe2aa"],"text":"A156992 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A156992 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}},{"id":"l-81cf8ab131866940","kind":"fingerprint_related","node_a_id":"n-62c0157c67ca5559","node_b_id":"n-88948f8cc95a6e83","fingerprint":"f6-9be755efdd908b94","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9be755efdd908b94"},"other_id":"n-62c0157c67ca5559","other":{"id":"n-62c0157c67ca5559","name":"theta_plus_one of Triangle read by rows: 2-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-9be755efdd908b94","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["4","10","3"]],"card_rows":[["1"],["2","2"],["4","10","3"],["8","38","27","4"]],"row_sums_preview":["1","4","17","77","372","1915","10481","60814"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6aaec68d653cbdb0","steps":["\u03b8+1"],"edges":["e-8098d1403de01053"],"text":"A143494 \u2192 \u03b8+1"},"headline":"A143494 \u2192 \u03b8+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2,   2;\n   3:  3,  10,    4;\n   4:  4,  27,   38,     8;\n   5:  5,  56,  165,   130,     16;\n   6:  6, 100,  500,   855,    422,     32;\n   7:  7, 162, 1225,  3640,   4053,   1330,     64;\n   8:  8, 245, 2604, 11900,  23604,  18207,   4118,    128;\n   9:  9, 352, 4998, 32508, 100905, 142296,  79005,  12610,   256;\n  10: 10, 486, 8880, 77910, 348642, 783300, 816820, 334935, 38342, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+(n-1)*x+(n-1)*x^2)*P(n-2,x) + (x^2+x^3)*d/dx P(n-2,x) + (2+(n+1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(1 + (n - 1)*x + (n - 1)*x^2)*P[n - 2, x] + (x^2 + x^3)*D[P[n - 2, x], x] + (2 + (n + 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = -(1 + (n - 1)*x + (n - 1)*x^2)*v[i - 2] + (x^2 + x^3)*deriv(v[i - 2], x) + (2 + (n + 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6aaec68d653cbdb0","steps":["\u03b8+1","reverse"],"edges":["e-8098d1403de01053","e-86990400404fca4b"],"text":"A143494 \u2192 \u03b8+1 \u2192 reverse"},"headline":"A143494 \u2192 \u03b8+1 \u2192 reverse","chain":null}