{"id":"n-b96b9868e74ecc25","revision":0,"first_index":1,"fingerprint":"f6-8ede4fc391b0c029","name":"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":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["2"],["8","8"],["46","72","24"],["352","688","384","64"],["3378","7600","5520","1600","160"],["39048","97112","83520","32320","5760","384"],["528414","1418648","1377096","642880","155680","18816","896"],["8196480","23393376","24912384","13315456","3942400","655872","57344","2048"],["143394210","430473312","493690656","292795776","100309440","20611584","2505216","165888","4608"],["2793408840","8752569768","10671542400","6879628160","2637868800","632362752","95692800","8878080","460800","10240"],["59971043790","194977600488","250360099704","172929637760","72594315200","19610502912","3485061888","405166080","29652480","1239040","22528"],["1406832628320","4724368986384","6343215094656","4645008601024","2101993344000","625267923456","125914263552","17284675584","1593630720","94392320","3244032","49152"],["35803283994450","123736555172880","172753484325552","133040455278016","64162355102560","20676482112000","4606815076864","719920920576","78731288064","5901209600","288763904","8306688","106496"],["982500375011400","3484100125645560","5035553742308544","4052768250707904","2064984725964160","712254669270144","172629132003328","29967727888384","3745566849024","334291077120","20779286528","854310912","20873216","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","16","142","1488","18258","258144","4142430","74475360","1483950690","32472422640","774364341870","19991576833200","555584281657650","16538860261425600"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["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":184322,"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-de507699fdfe774e"]},{"id":184323,"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-de507699fdfe774e"]},{"id":184324,"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-de507699fdfe774e"]},{"id":184325,"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-de507699fdfe774e"]}],"region_exact":{},"incoming_edges":[{"id":"e-de507699fdfe774e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-b96b9868e74ecc25","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":"c97903a5bdc980f6b41dfb63f3424d5fa614582bf1e6b19a4bf4eb13ab5e526a","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-2f5fe77ca4236d1d","kind":"fingerprint_related","node_a_id":"n-b96b9868e74ecc25","node_b_id":"n-daf465941b3d6501","fingerprint":"f6-8ede4fc391b0c029","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ede4fc391b0c029"},"other_id":"n-daf465941b3d6501","other":{"id":"n-daf465941b3d6501","name":"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":"f6-8ede4fc391b0c029","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["32","32"]],"card_rows":[["4"],["32","32"],["368","576","192"],["5632","11008","6144","1024"]],"row_sums_preview":["0","4","64","1136","23808","584256","16521216","530231040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013611","D"],"edges":["e-403803ced55fc794","e-5d74b160ceaaef10"],"text":"A290319 \u2192 basis A013611 \u2192 D"},"headline":"A290319 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-4e5e64d8e71c5486","kind":"fingerprint_related","node_a_id":"n-b96b9868e74ecc25","node_b_id":"n-351bdce99219d68c","fingerprint":"f6-8ede4fc391b0c029","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8ede4fc391b0c029"},"other_id":"n-351bdce99219d68c","other":{"id":"n-351bdce99219d68c","name":"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":"f6-8ede4fc391b0c029","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["24","24"]],"card_rows":[["2"],["24","24"],["690","1080","360"],["36960","72240","40320","6720"]],"row_sums_preview":["0","2","48","2130","156240","17253810","2683406880","559787278050"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A013609","D"],"edges":["e-2613ba1fbc2604f1","e-feb2ae34493b08d1"],"text":"A204420 \u2192 basis A013609 \u2192 D"},"headline":"A204420 \u2192 basis A013609 \u2192 D","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:         0;\n   2:         2;\n   3:         8,         8;\n   4:        46,        72,        24;\n   5:       352,       688,       384,        64;\n   6:      3378,      7600,      5520,      1600,       160;\n   7:     39048,     97112,     83520,     32320,      5760,      384;\n   8:    528414,   1418648,   1377096,    642880,    155680,    18816,     896;\n   9:   8196480,  23393376,  24912384,  13315456,   3942400,   655872,   57344,   2048;\n  10: 143394210, 430473312, 493690656, 292795776, 100309440, 20611584, 2505216, 165888, 4608;\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) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\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] = 0; v[2] = 2; 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"],"edges":["e-de507699fdfe774e"],"text":"A161198 \u2192 D"},"headline":"A161198 \u2192 D","chain":null}