{"id":"n-66b5fd0d041a4c11","revision":0,"first_index":1,"fingerprint":"f6-236e6d52004bfa9f","name":"one_plus_derivative of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","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":true,"rows":[["9","40","16"],["13","108","240","64"],["17","208","864","1280","256"],["21","340","2080","5760","6400","1024"],["25","504","4080","16640","34560","30720","4096"],["29","700","7056","38080","116480","193536","143360","16384"],["33","928","11200","75264","304640","745472","1032192","655360","65536"],["37","1188","16704","134400","677376","2193408","4472832","5308416","2949120","262144"],["41","1480","23760","222720","1344000","5419008","14622720","25559040","26542080","13107200","1048576"],["45","1804","32560","348480","2449920","11827200","39739392","91914240","140574720","129761280","57671680","4194304"],["49","2160","43296","520960","4181760","23519232","94617600","272498688","551485440","749731840","622854144","251658240","16777216"],["53","2548","56160","750464","6772480","43490304","203833344","702873600","1771241472","3186360320","3898605568","2944401408","1090519040","67108864"],["57","2968","71344","1048320","10506496","75851776","405909504","1630666752","4920115200","11021058048","17843617792","19847446528","13740539904","4697620480","268435456"],["61","3420","89040","1426880","15724800","126077952","758517760","3479224320","12230000640","32800768000","66126348288","97328824320","99237232640","63417876480","20132659200","1073741824"],["65","3904","109440","1899520","22830080","201277440","1344831488","6935019520","27833794560","86968893440","209924915200","384735117312","519087063040","488552529920","289910292480","85899345920","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["65","425","2625","15625","90625","515625","2890625","16015625","87890625","478515625","2587890625","13916015625","74462890625","396728515625","2105712890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["9","40","16"]],"recurrence":{"denominator":{"coeffs":[["1","4"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["5","24","16"],["4","16","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182539,"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-e5ed55f6b6fce737"]},{"id":182540,"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-e5ed55f6b6fce737"]},{"id":182541,"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-e5ed55f6b6fce737"]},{"id":182542,"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-e5ed55f6b6fce737"]}],"region_exact":{},"incoming_edges":[{"id":"e-e5ed55f6b6fce737","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-66b5fd0d041a4c11","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":"2c2e4cb0fa3f5c40b9936c90ba6aebd4d89e58fcafa656fd04cadcc502a217b0","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-146f1cd501c36740","role":"sequence","name":"basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","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-1e7c72b02d7f592c","kind":"fingerprint_related","node_a_id":"n-369cec4fbc46cd80","node_b_id":"n-66b5fd0d041a4c11","fingerprint":"f6-236e6d52004bfa9f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-236e6d52004bfa9f"},"other_id":"n-369cec4fbc46cd80","other":{"id":"n-369cec4fbc46cd80","name":"one_plus_derivative of derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-236e6d52004bfa9f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["40","32"]],"card_rows":[["4"],["40","32"],["108","480","192"],["208","1728","3840","1024"]],"row_sums_preview":["0","4","72","780","6800","52500","375000","2537500"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D","D+1"],"edges":["e-b04b02754d5cebbf","e-c14c2a09f1ebead5"],"text":"A013611 \u2192 D \u2192 D+1"},"headline":"A013611 \u2192 D \u2192 D+1","chain":null}},{"id":"l-2883be84be4cbaec","kind":"fingerprint_related","node_a_id":"n-bf9d0f008400b3c6","node_b_id":"n-66b5fd0d041a4c11","fingerprint":"f6-236e6d52004bfa9f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-236e6d52004bfa9f"},"other_id":"n-bf9d0f008400b3c6","other":{"id":"n-bf9d0f008400b3c6","name":"one_plus_derivative of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-236e6d52004bfa9f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","4"],["9","40","16"]],"card_rows":[["1"],["5","4"],["9","40","16"],["13","108","240","64"]],"row_sums_preview":["1","9","65","425","2625","15625","90625","515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2e0409c0014ec240","steps":["D+1"],"edges":["e-5d20fd95bc57193f"],"text":"A013611 \u2192 D+1"},"headline":"A013611 \u2192 D+1","chain":null}},{"id":"l-8fc7873aed563c56","kind":"fingerprint_related","node_a_id":"n-66b5fd0d041a4c11","node_b_id":"n-716a133f6399cf18","fingerprint":"f6-236e6d52004bfa9f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-236e6d52004bfa9f"},"other_id":"n-716a133f6399cf18","other":{"id":"n-716a133f6399cf18","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-236e6d52004bfa9f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","4"],["18","80","32"]],"card_rows":[["1"],["5","4"],["18","80","32"],["78","648","1440","384"]],"row_sums_preview":["1","9","130","2550","63000","1875000","65250000","2598750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","D+1"],"edges":["e-3656191e13afb7f3","e-125b2891817ff4ec"],"text":"A136572 \u2192 basis A013611 \u2192 D+1"},"headline":"A136572 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-f3530d651eae0b42","kind":"fingerprint_related","node_a_id":"n-c41e614167a72a8f","node_b_id":"n-66b5fd0d041a4c11","fingerprint":"f6-236e6d52004bfa9f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-236e6d52004bfa9f"},"other_id":"n-c41e614167a72a8f","other":{"id":"n-c41e614167a72a8f","name":"one_plus_derivative of basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-236e6d52004bfa9f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","4"],["9","40","16"]],"card_rows":[["1"],["5","4"],["9","40","16"],["13","108","240","64"]],"row_sums_preview":["1","9","65","425","2625","15625","90625","515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","D+1"],"edges":["e-f93ad378f9476e41","e-917a93f0e9a79337"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-f5207e3026163a5d","kind":"fingerprint_related","node_a_id":"n-66b5fd0d041a4c11","node_b_id":"n-8a1a31d1b19d37ec","fingerprint":"f6-236e6d52004bfa9f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-236e6d52004bfa9f"},"other_id":"n-8a1a31d1b19d37ec","other":{"id":"n-8a1a31d1b19d37ec","name":"one_plus_derivative of basis_map_A013611 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-236e6d52004bfa9f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","8"],["36","160","64"]],"card_rows":[["1"],["10","8"],["36","160","64"],["104","864","1920","512"]],"row_sums_preview":["1","18","260","3400","42000","500000","5800000","66000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","D+1"],"edges":["e-d7d2422771947afc","e-b29cc92b92be4d0c"],"text":"A130123 \u2192 basis A013611 \u2192 D+1"},"headline":"A130123 \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:  9,   40,    16;\n   2: 13,  108,   240,     64;\n   3: 17,  208,   864,   1280,     256;\n   4: 21,  340,  2080,   5760,    6400,     1024;\n   5: 25,  504,  4080,  16640,   34560,    30720,     4096;\n   6: 29,  700,  7056,  38080,  116480,   193536,   143360,    16384;\n   7: 33,  928, 11200,  75264,  304640,   745472,  1032192,   655360,     65536;\n   8: 37, 1188, 16704, 134400,  677376,  2193408,  4472832,  5308416,   2949120,    262144;\n   9: 41, 1480, 23760, 222720, 1344000,  5419008, 14622720, 25559040,  26542080,  13107200,  1048576;\n  10: 45, 1804, 32560, 348480, 2449920, 11827200, 39739392, 91914240, 140574720, 129761280, 57671680, 4194304;\n","recurrence_sentence":"The row polynomials satisfy (4*n+1+4*x)*P(n,x) = (4*n+5+(16*n+24)*x+16*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 9 + 40*x + 16*x^2.","mathematica":"P[1, x_] := 9 + 40*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 5 + (16*n + 24)*x + 16*x^2)*P[n - 1, x])/(4*n + 1 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 9 + 40*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = ((4*n + 5 + (16*n + 24)*x + 16*x^2)*v[i - 1])/(4*n + 1 + 4*x)); 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-5fe240122e98ffb3","steps":["basis A013611","D+1"],"edges":["e-b91bf4c52fd06745","e-e5ed55f6b6fce737"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 D+1"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 D+1","chain":null}