{"id":"n-81977002e81a543b","revision":0,"first_index":0,"fingerprint":"f6-54feb342b7863b00","name":"one_plus_derivative of basis_map_A013609 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.","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"],["4","2"],["28","24","4"],["304","328","96","8"],["4496","5568","2144","320","16"],["84288","114848","52480","10560","960","32"],["1914816","2803328","1445056","352000","43840","2688","64"],["51115776","79215232","44731904","12487552","1908480","162176","7168","128"],["1568067840","2546325504","1545784320","478779392","84801024","8931328","551936","18432","256"],["54352143360","91809501696","59146493952","19901532160","3947317248","485882880","37502976","1763328","46080","512"],["2100667898880","3670883186688","2486286357504","895615344640","194485852160","27000164352","2451843072","144875520","5360640","112640","1024"],["89568936898560","161238880290816","114006451249152","43499055474688","10179784581120","1556743327744","160350339072","11207184384","523776000","15656960","270336","2048"],["4176488273448960","7718947064856576","5666603049492480","2271697578016768","566335698497536","93828176707584","10698780950528","852162871296","47318396928","1794129920","44244992","638976","4096"],["211415003773747200","400062716235325440","303605800446394368","127083235752132608","33454741181841408","5931702291587072","735934449975296","65156921589760","4139384291328","187176411136","5876023296","121618432","1490944","8192"],["11545939708847308800","22319517186737602560","17448251121570004992","7587639376807526400","2094780851874873344","393791631271886848","52486061451952128","5074581570060288","359599962275840","18666228252672","701100015616","18529452032","326516736","3440640","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","56","736","12544","263168","6561792","189628416","6233260032","229682184192","9377540997120","420221949050880","20505499942256640","1082358623612436480","61447859506281185280"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-36","24","-4"],["48","-16","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","4"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188688,"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-3bb987537a007943"]},{"id":188689,"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-3bb987537a007943"]},{"id":188690,"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-3bb987537a007943"]},{"id":188691,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-3bb987537a007943"]}],"region_exact":{},"incoming_edges":[{"id":"e-3bb987537a007943","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-81977002e81a543b","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":"6d1100445a7ef32befd93ca05737855eef4c80b0d529655984117d1536bc6f27","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-de50e58c7c589630","role":"sequence","name":"basis_map_A013609 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.","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-1573f951ef9d84e9","kind":"fingerprint_related","node_a_id":"n-81977002e81a543b","node_b_id":"n-2fc865e4fc7ead86","fingerprint":"f6-54feb342b7863b00","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-54feb342b7863b00"},"other_id":"n-2fc865e4fc7ead86","other":{"id":"n-2fc865e4fc7ead86","name":"one_plus_derivative of basis_map_A007318 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":"f6-54feb342b7863b00","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["21","18","3"]],"card_rows":[["1"],["2","1"],["21","18","3"],["570","615","180","15"]],"row_sums_preview":["1","3","42","1380","82320","7771680","1065778560","200198718720"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1336d058dca9979e","steps":["basis A007318","D+1"],"edges":["e-857dad5fd144a15f","e-6e1f16e576c9b556"],"text":"A204420 \u2192 basis A007318 \u2192 D+1"},"headline":"A204420 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-9a0a0aa326f64869","kind":"fingerprint_related","node_a_id":"n-91ac48d9545960c8","node_b_id":"n-81977002e81a543b","fingerprint":"f6-54feb342b7863b00","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-54feb342b7863b00"},"other_id":"n-91ac48d9545960c8","other":{"id":"n-91ac48d9545960c8","name":"one_plus_derivative of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-54feb342b7863b00","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["7","6","1"]],"card_rows":[["1"],["2","1"],["7","6","1"],["38","41","12","1"]],"row_sums_preview":["1","3","14","92","784","8224","102528","1481472"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["D+1"],"edges":["e-28c8bc40e3e47ff1"],"text":"A028338 \u2192 D+1"},"headline":"A028338 \u2192 D+1","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:           4,           2;\n  2:          28,          24,           4;\n  3:         304,         328,          96,           8;\n  4:        4496,        5568,        2144,         320,         16;\n  5:       84288,      114848,       52480,       10560,        960,        32;\n  6:     1914816,     2803328,     1445056,      352000,      43840,      2688,       64;\n  7:    51115776,    79215232,    44731904,    12487552,    1908480,    162176,     7168,     128;\n  8:  1568067840,  2546325504,  1545784320,   478779392,   84801024,   8931328,   551936,   18432,   256;\n  9: 54352143360, 91809501696, 59146493952, 19901532160, 3947317248, 485882880, 37502976, 1763328, 46080, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-48*n+36+(16*n-24)*x+4*x^2)*P(n-2,x) + (8*n-8+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 48*n + 36 + (16*n - 24)*x + 4*x^2)*P[n - 2, x] + (8*n - 8 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 48*n + 36 + (16*n - 24)*x + 4*x^2)*v[i - 2] + (8*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4172472abe953cea","steps":["basis A013609","D+1"],"edges":["e-294a6ed01a588998","e-3bb987537a007943"],"text":"A290319 \u2192 basis A013609 \u2192 D+1"},"headline":"A290319 \u2192 basis A013609 \u2192 D+1","chain":null}
