{"id":"n-07593db20f25af3f","revision":0,"first_index":1,"fingerprint":"f6-42e9f6f4286ebba9","name":"one_plus_derivative of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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"],["3","2"],["11","16","4"],["61","118","60","8"],["457","1040","728","192","16"],["4323","10978","9320","3440","560","32"],["49443","136160","132076","60160","13840","1536","64"],["663549","1947062","2086420","1101912","316400","49952","4032","128"],["10223505","31589856","36609072","21619584","7271264","1444352","166656","10240","256"],["177853635","573867522","708927312","457359328","173508384","40673472","5940480","523776","25344","512"],["3448137915","11545978608","15047827284","10436808960","4357775840","1159936512","201086592","22548480","1570560","61440","1024"],["73720354365","254948644278","347848899948","256383004328","115826724640","34129365952","6753479040","903032064","80298240","4533760","146432","2048"],["1723066771545","6131201614704","8705399587848","6759413632576","3263245494256","1045666592256","230125584128","35272427520","3754215168","271462400","12683264","344064","4096"],["43709137575075","159539839167330","234621761911992","190624950053200","97422468922064","33508953132512","8052895428864","1378037360128","168721403136","14649130496","878884864","34557952","798720","8192"],["1195958421688275","4466600500656960","6777603805131324","5731286164810752","3078176788641136","1125251614462976","291338243548352","54629032460288","7491532835072","750465171456","54208394240","2743336960","92065792","1835008","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","31","247","2433","28653","393279","6169455","108934785","2138679765","46221733215","1090598485095","27897430413825","769042328334525","22729143615054975"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-25","20","-4"],["20","-8","0"],["-4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","4"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184326,"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-632365d14fbacf91"]},{"id":184327,"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-632365d14fbacf91"]},{"id":184328,"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-632365d14fbacf91"]},{"id":184329,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-632365d14fbacf91"]}],"region_exact":{},"incoming_edges":[{"id":"e-632365d14fbacf91","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-07593db20f25af3f","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"e9e1b21ad7aca01f0aba651e7167b536210ff93c79c9f5e12184c49421fa86bb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7c9c430277778c10","role":"sequence","name":"Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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-2985b707f59cbee4","kind":"fingerprint_related","node_a_id":"n-07593db20f25af3f","node_b_id":"n-6577090cee20b8ba","fingerprint":"f6-42e9f6f4286ebba9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42e9f6f4286ebba9"},"other_id":"n-6577090cee20b8ba","other":{"id":"n-6577090cee20b8ba","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","first_index":0,"fingerprint":"f6-42e9f6f4286ebba9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["33","48","12"]],"card_rows":[["1"],["3","2"],["33","48","12"],["915","1770","900","120"]],"row_sums_preview":["1","5","93","3705","255465","27077085","4088135205","833709301425"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013609","D+1"],"edges":["e-2613ba1fbc2604f1","e-dbfa2177d006c2e3"],"text":"A204420 \u2192 basis A013609 \u2192 D+1"},"headline":"A204420 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-b3f8cd70386641ae","kind":"fingerprint_related","node_a_id":"n-07593db20f25af3f","node_b_id":"n-5ca0512feb069e44","fingerprint":"f6-42e9f6f4286ebba9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-42e9f6f4286ebba9"},"other_id":"n-5ca0512feb069e44","other":{"id":"n-5ca0512feb069e44","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","first_index":0,"fingerprint":"f6-42e9f6f4286ebba9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["44","64","16"]],"card_rows":[["1"],["6","4"],["44","64","16"],["488","944","480","64"]],"row_sums_preview":["1","10","124","1976","38928","916896","25169856","789690240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013611","D+1"],"edges":["e-403803ced55fc794","e-7b582e6cac0ad202"],"text":"A290319 \u2192 basis A013611 \u2192 D+1"},"headline":"A290319 \u2192 basis A013611 \u2192 D+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:         3,         2;\n   3:        11,        16,         4;\n   4:        61,       118,        60,         8;\n   5:       457,      1040,       728,       192,        16;\n   6:      4323,     10978,      9320,      3440,       560,       32;\n   7:     49443,    136160,    132076,     60160,     13840,     1536,      64;\n   8:    663549,   1947062,   2086420,   1101912,    316400,    49952,    4032,    128;\n   9:  10223505,  31589856,  36609072,  21619584,   7271264,  1444352,  166656,  10240,   256;\n  10: 177853635, 573867522, 708927312, 457359328, 173508384, 40673472, 5940480, 523776, 25344, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-20*n+25+(8*n-20)*x+4*x^2)*P(n-2,x) + (4*n-8+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*P[n - 2, x] + (4*n - 8 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*v[i - 2] + (4*n - 8 + 4*x)*v[i - 1]); 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-7c9c430277778c10","steps":["D+1"],"edges":["e-632365d14fbacf91"],"text":"A161198 \u2192 D+1"},"headline":"A161198 \u2192 D+1","chain":null}
