{"id":"n-41fdaa260044e0da","revision":0,"first_index":1,"fingerprint":"f6-882d99bdfc839db5","name":"reversal of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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"],["4","3"],["16","24","8"],["64","144","100","21"],["256","768","816","360","55"],["1024","3840","5504","3744","1200","144"],["4096","18432","33280","30720","15216","3816","377"],["16384","86016","187392","218880","147520","57168","11756","987"],["65536","393216","1003520","1419264","1213440","640512","203312","35400","2584"],["262144","1769472","5177344","8601600","8924160","5983488","2587264","694368","104780","6765"],["1048576","7864320","25952256","49545216","60497920","49287168","27086080","9898752","2298912","306000","17711"],["4194304","34603008","127139840","274268160","385449984","370040832","247281664","114869760","36304640","7426368","884064","46368"],["16777216","150994944","611319808","1470627840","2338652160","2586968064","2038751232","1152036864","462741760","128720640","23516448","2531664","121393"],["67108864","654311424","2894069760","7681867776","13639352320","17089560576","15530459136","10354360320","5059795968","1788238080","443923072","73249632","7196500","317811"],["268435456","2818572288","13522436096","39258685440","76991692800","107805671424","111051866112","85421850624","49263513600","21180303360","6677664768","1496047104","225018448","20329080","832040"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","48","329","2255","15456","105937","726103","4976784","34111385","233802911","1602508992","10983760033","75283811239","516002918640"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-8"],["8"]]},"terms":[{"coeff":{"coeffs":[["-32","-48","-10"],["32","36","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-16","-24","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181981,"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-932847c63aff17e7"]},{"id":181982,"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-932847c63aff17e7"]},{"id":181983,"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-932847c63aff17e7"]}],"region_exact":{},"incoming_edges":[{"id":"e-932847c63aff17e7","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-41fdaa260044e0da","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":"0383a040d33a0cdf63d164b7e110e9808921acd173531b8b15c77092470c6629","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-02f1a3354aa99430","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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-259733479d339d57","kind":"fingerprint_related","node_a_id":"n-02f1a3354aa99430","node_b_id":"n-41fdaa260044e0da","fingerprint":"f6-882d99bdfc839db5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-882d99bdfc839db5"},"other_id":"n-02f1a3354aa99430","other":{"id":"n-02f1a3354aa99430","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","first_index":1,"fingerprint":"f6-882d99bdfc839db5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["8","24","16"]],"card_rows":[["1"],["3","4"],["8","24","16"],["21","100","144","64"]],"row_sums_preview":["1","7","48","329","2255","15456","105937","726103"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["basis A013611"],"edges":["e-60bc1ac61eca3352"],"text":"A078812 \u2192 basis A013611"},"headline":"A078812 \u2192 basis A013611","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:      4,       3;\n   3:     16,      24,       8;\n   4:     64,     144,     100,      21;\n   5:    256,     768,     816,     360,      55;\n   6:   1024,    3840,    5504,    3744,    1200,     144;\n   7:   4096,   18432,   33280,   30720,   15216,    3816,     377;\n   8:  16384,   86016,  187392,  218880,  147520,   57168,   11756,    987;\n   9:  65536,  393216, 1003520, 1419264, 1213440,  640512,  203312,  35400,   2584;\n  10: 262144, 1769472, 5177344, 8601600, 8924160, 5983488, 2587264, 694368, 104780, 6765;\n","recurrence_sentence":"The row polynomials satisfy (8*n-8)*P(n,x) = (32*n-32+(36*n-48)*x+(5*n-10)*x^2)*P(n-1,x) - (16*x+24*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n - 32 + (36*n - 48)*x + (5*n - 10)*x^2)*P[n - 1, x] - (16*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(8*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n - 32 + (36*n - 48)*x + (5*n - 10)*x^2)*v[i - 1] - (16*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(8*n - 8)); 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-976cdb6e4a693ec8","steps":["basis A013611","reverse"],"edges":["e-60bc1ac61eca3352","e-932847c63aff17e7"],"text":"A078812 \u2192 basis A013611 \u2192 reverse"},"headline":"A078812 \u2192 basis A013611 \u2192 reverse","chain":null}