{"id":"n-31eb1382c3434f27","revision":0,"first_index":0,"fingerprint":"f6-1e7077c59c0f616e","name":"reversal of The convolution matrix of the double factorial of odd numbers (A001147).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","3"],["1","9","15"],["1","18","87","105"],["1","30","285","975","945"],["1","45","705","4680","12645","10395"],["1","63","1470","15960","82845","187425","135135"],["1","84","2730","43890","370125","1595790","3133935","2027025"],["1","108","4662","103950","1291815","8998290","33453945","58437855","34459425"],["1","135","7470","220500","3786615","38710035","231416325","760970700","1203216525","654729075"],["1","165","11385","429660","9740115","137162025","1199167200","6314962500","18705542625","27125492625","13749310575"],["1","198","16665","782595","22629915","419625360","5039703900","38692476900","182894567625","494764058250","664761133575","316234143225"],["1","234","23595","1349205","48455550","1143782640","18048090060","189883593900","1304976397725","5615811951750","14023390706325","17600023616175","7905853580625"],["1","273","32487","2222220","97026930","2839726890","56936699820","785470565880","7381251402525","46070080281225","182497749258825","424278354099600","500706514833525","213458046676875"],["1","315","43680","3521700","183693510","6526750230","162006594750","2835191759400","34872721208325","297032627967075","1702810398539250","6264206330382000","13653335491921125","15234653491682625","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","25","211","2236","28471","422899","7173580","136750051","2893057381","67241818876","1702829138209","46659181547785","1375237342827076","43380198327693361"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","3"]]},"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":186198,"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-95aec81e5f9bc7b2"]},{"id":186199,"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-95aec81e5f9bc7b2"]},{"id":186200,"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-95aec81e5f9bc7b2"]}],"region_exact":{},"incoming_edges":[{"id":"e-95aec81e5f9bc7b2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-31eb1382c3434f27","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":"bf415a9475c7cab47c8b2a02b96120c8c1f8c3a0d7f0930b7459bca24260a9c7","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-ca41bee2bc8fcdbe","role":"sequence","name":"The convolution matrix of the double factorial of odd numbers (A001147).","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-e7b0670fc125e8f8","kind":"fingerprint_related","node_a_id":"n-31eb1382c3434f27","node_b_id":"n-307dc0a9b33c92f9","fingerprint":"f6-1e7077c59c0f616e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e7077c59c0f616e"},"other_id":"n-307dc0a9b33c92f9","other":{"id":"n-307dc0a9b33c92f9","name":"basis_map_A008277 of basis_map_A013609 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-1e7077c59c0f616e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","2"],["60","36","4"]],"card_rows":[["1"],["6","2"],["60","36","4"],["840","696","144","8"]],"row_sums_preview":["1","8","100","1688","35776","911072","27065536","918218240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013609","basis A008277"],"edges":["e-8d6d917622b49386","e-c549735663fe9b4e"],"text":"A225471 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A225471 \u2192 basis A013609 \u2192 basis A008277","chain":null}},{"id":"l-fdc2c3f03598914e","kind":"fingerprint_related","node_a_id":"n-ca41bee2bc8fcdbe","node_b_id":"n-31eb1382c3434f27","fingerprint":"f6-1e7077c59c0f616e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e7077c59c0f616e"},"other_id":"n-ca41bee2bc8fcdbe","other":{"id":"n-ca41bee2bc8fcdbe","name":"The convolution matrix of the double factorial of odd numbers (A001147).","first_index":0,"fingerprint":"f6-1e7077c59c0f616e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["15","9","1"]],"card_rows":[["1"],["3","1"],["15","9","1"],["105","87","18","1"]],"row_sums_preview":["1","4","25","211","2236","28471","422899","7173580"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A035342"],"derivation":{"root":"n-ca41bee2bc8fcdbe","steps":[],"edges":[],"text":"A035342"},"headline":"A035342: The convolution matrix of the double factorial of odd numbers (A001147).","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,   3;\n  2: 1,   9,   15;\n  3: 1,  18,   87,    105;\n  4: 1,  30,  285,    975,     945;\n  5: 1,  45,  705,   4680,   12645,    10395;\n  6: 1,  63, 1470,  15960,   82845,   187425,    135135;\n  7: 1,  84, 2730,  43890,  370125,  1595790,   3133935,   2027025;\n  8: 1, 108, 4662, 103950, 1291815,  8998290,  33453945,  58437855,   34459425;\n  9: 1, 135, 7470, 220500, 3786615, 38710035, 231416325, 760970700, 1203216525, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*n*x)*P(n-1,x) - x^2*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[(1 + 3*n*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\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] = (1 + 3*n*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ca41bee2bc8fcdbe","steps":["reverse"],"edges":["e-95aec81e5f9bc7b2"],"text":"A035342 \u2192 reverse"},"headline":"A035342 \u2192 reverse","chain":null}