{"id":"n-716a133f6399cf18","revision":0,"first_index":0,"fingerprint":"f6-236e6d52004bfa9f","name":"one_plus_derivative of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","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":true,"rows":[["1"],["5","4"],["18","80","32"],["78","648","1440","384"],["408","4992","20736","30720","6144"],["2520","40800","249600","691200","768000","122880"],["18000","362880","2937600","11980800","24883200","22118400","2949120"],["146160","3528000","35562240","191923200","587059200","975421440","722534400","82575360"],["1330560","37416960","451584000","3034644480","12283084800","30057431040","41617981440","26424115200","2642411520"],["13426560","431101440","6061547520","48771072000","245806202880","795943895040","1623101276160","1926317998080","1070176665600","95126814720"],["148780800","5370624000","86220288000","808206336000","4877107200000","19664496230400","53062926336000","92748644352000","96315899904000","47563407360000","3805072588800"],["1796256000","72009907200","1299691008000","13910206464000","97792966656000","472103976960000","1586269362585600","3668922335232000","5611292983296000","5179655061504000","2302068916224000","167423193907200"],["23471078400","1034643456000","20738853273600","249540673536000","2003069730816000","11265749758771200","45321981788160000","130527307549900800","264162408136704000","359122750930944000","298348131542630400","120544699613184000","8036313307545600"],["330032102400","15866448998400","349709488128000","4673154937651200","42172373827584000","270815027606323200","1269274472821555200","4376808526970880000","11029557487966617600","19841531988934656000","24276697962931814400","18334848811165286400","6790684744876032000","417888291992371200"],["4969162598400","258745168281600","6219648007372800","91390746230784000","915938367779635200","6612628216165171200","35386496940559564800","142158740956014182400","428927235643146240000","960797007840647577600","1555576107932477030400","1730266472994412953600","1197876788996132044800","409530526152523776000","23401744351572787200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","130","2550","63000","1875000","65250000","2598750000","116550000000","5811750000000","318937500000000","19100812500000000","1239603750000000000","86655318750000000000","6491547562500000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-3","4"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","8","16"],["4","16","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190551,"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-125b2891817ff4ec"]},{"id":190552,"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-125b2891817ff4ec"]},{"id":190553,"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-125b2891817ff4ec"]},{"id":190554,"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-125b2891817ff4ec"]}],"region_exact":{},"incoming_edges":[{"id":"e-125b2891817ff4ec","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-716a133f6399cf18","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":"6a0476bc43f11747ca98eb3e842ca935709d215d77032dd675904f9139e70938","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-872d1ad8224c1da6","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","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-218676d068aecddb","kind":"fingerprint_related","node_a_id":"n-c41e614167a72a8f","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-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-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-66b5fd0d041a4c11","other":{"id":"n-66b5fd0d041a4c11","name":"one_plus_derivative of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","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":[["9","40","16"],["13","108","240","64"],["17","208","864","1280","256"]],"card_rows":[["9","40","16"],["13","108","240","64"],["17","208","864","1280","256"],["21","340","2080","5760","6400","1024"]],"row_sums_preview":["65","425","2625","15625","90625","515625","2890625","16015625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-9f3a5d169410347d","kind":"fingerprint_related","node_a_id":"n-bf9d0f008400b3c6","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-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-cc4d2f0358d2bd74","kind":"fingerprint_related","node_a_id":"n-8a1a31d1b19d37ec","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-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}},{"id":"l-cd82a14750a622bd","kind":"fingerprint_related","node_a_id":"n-369cec4fbc46cd80","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-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}}],"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:        5,         4;\n  2:       18,        80,         32;\n  3:       78,       648,       1440,         384;\n  4:      408,      4992,      20736,       30720,         6144;\n  5:     2520,     40800,     249600,      691200,       768000,       122880;\n  6:    18000,    362880,    2937600,    11980800,     24883200,     22118400,       2949120;\n  7:   146160,   3528000,   35562240,   191923200,    587059200,    975421440,     722534400,      82575360;\n  8:  1330560,  37416960,  451584000,  3034644480,  12283084800,  30057431040,   41617981440,   26424115200,    2642411520;\n  9: 13426560, 431101440, 6061547520, 48771072000, 245806202880, 795943895040, 1623101276160, 1926317998080, 1070176665600, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy (4*n-3+4*x)*P(n,x) = (4*n^2+n+(16*n^2+8*n)*x+16*n*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + n + (16*n^2 + 8*n)*x + 16*n*x^2)*P[n - 1, x])/(4*n - 3 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + n + (16*n^2 + 8*n)*x + 16*n*x^2)*v[i - 1])/(4*n - 3 + 4*x)); 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-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}