{"id":"n-c5989f7729c933c8","revision":0,"first_index":0,"fingerprint":"f6-640a8dc8403e6c62","name":"derivative of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","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":[[],["20"],["840","800"],["26460","50400","24000"],["740880","2116800","2016000","640000"],["19448100","74088000","105840000","67200000","16000000"],["490092120","2333772000","4445280000","4233600000","2016000000","384000000"],["12007256940","68612896800","163364040000","207446400000","148176000000","56448000000","8960000000"],["288174166560","1921161110400","5489031744000","8712748800000","8297856000000","4741632000000","1505280000000","204800000000"],["6808114684980","51871349980800","172904499936000","329341904640000","392073696000000","298722816000000","142248960000000","38707200000000","4608000000000"],["158856009316200","1361622936996000","5187134998080000","11526966662400000","16467095232000000","15682947840000000","9957427200000000","4064256000000000","967680000000000","102400000000000"],["3669573815204220","34948322049564000","149778523069560000","380389899859200000","633983166432000000","724552190208000000","575041420800000000","312947712000000000","111767040000000000","23654400000000000","2252800000000000"],["84066600130133040","880697715649012800","4193798645947680000","11982281845564800000","22823393991552000000","30431191988736000000","28982087608320000000","19715705856000000000","9388431360000000000","2980454400000000000","567705600000000000","49152000000000000"],["1912515152960526660","21857316033834590400","114490703034371664000","363462549315465600000","778848319961712000000","1186816487560704000000","1318684986178560000000","1076477539737600000000","640760440320000000000","271221350400000000000","77491814400000000000","13418496000000000000","1064960000000000000"],["43252265766953449080","535504242828947464800","3060024244736842656000","10685798949874688640000","25442378452082592000000","43615505917855872000000","55384769419499520000000","52747399447142400000000","37676713890816000000000","19934769254400000000000","7594197811200000000000","1972518912000000000000","313098240000000000000","22937600000000000000"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","20","1640","100860","5513680","282576100","13902744120","665014593740","31160683820960","1437286541241780","65476386878792200","2952985048233528220","132078967611899625840","5866507478095208381060","259028868648203816209880"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["20"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["21","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193144,"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-cf836392720f7b39"]},{"id":193145,"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-cf836392720f7b39"]},{"id":193146,"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-cf836392720f7b39"]},{"id":193147,"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-cf836392720f7b39"]}],"region_exact":{},"incoming_edges":[{"id":"e-cf836392720f7b39","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-c5989f7729c933c8","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":"489b837fbe40205234f7c75aafd55e6d4c0ca0da6ce25211d62ac3e13b02f21f","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-fbf962df53148c52","role":"sequence","name":"basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","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-813ed13ecfa9c595","kind":"fingerprint_related","node_a_id":"n-c5989f7729c933c8","node_b_id":"n-509590a43fbc65c5","fingerprint":"f6-640a8dc8403e6c62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-640a8dc8403e6c62"},"other_id":"n-509590a43fbc65c5","other":{"id":"n-509590a43fbc65c5","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","first_index":0,"fingerprint":"f6-640a8dc8403e6c62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1600","3360","1764"],["64000","201600","211680","74088"],["2560000","10752000","16934400","11854080","3111696"]],"card_rows":[["1600","3360","1764"],["64000","201600","211680","74088"],["2560000","10752000","16934400","11854080","3111696"],["102400000","537600000","1128960000","1185408000","622339200","130691232"]],"row_sums_preview":["6724","551368","45212176","3707398432","304006671424","24928547056768","2044140858654976","167619550409708032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5cf6acfdac607d6d","steps":["basis A013610","basis A013609"],"edges":["e-065a527a7a0e90c1","e-6372524cd1a6352e"],"text":"A038333 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038333 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-9623439fdbe12ad8","kind":"fingerprint_related","node_a_id":"n-c5989f7729c933c8","node_b_id":"n-10de7fc69fcc7b29","fingerprint":"f6-640a8dc8403e6c62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-640a8dc8403e6c62"},"other_id":"n-10de7fc69fcc7b29","other":{"id":"n-10de7fc69fcc7b29","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-640a8dc8403e6c62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1764","3360","1600"],["74088","211680","201600","64000"],["3111696","11854080","16934400","10752000","2560000"]],"card_rows":[["1764","3360","1600"],["74088","211680","201600","64000"],["3111696","11854080","16934400","10752000","2560000"],["130691232","622339200","1185408000","1128960000","537600000","102400000"]],"row_sums_preview":["6724","551368","45212176","3707398432","304006671424","24928547056768","2044140858654976","167619550409708032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["basis A013609","basis A013609"],"edges":["e-0c8cc5263f914382","e-db5897451ba89947"],"text":"A038336 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038336 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-96d01ef9db1ac6d9","kind":"fingerprint_related","node_a_id":"n-fbf962df53148c52","node_b_id":"n-c5989f7729c933c8","fingerprint":"f6-640a8dc8403e6c62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-640a8dc8403e6c62"},"other_id":"n-fbf962df53148c52","other":{"id":"n-fbf962df53148c52","name":"basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-640a8dc8403e6c62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","20"],["441","840","400"]],"card_rows":[["1"],["21","20"],["441","840","400"],["9261","26460","25200","8000"]],"row_sums_preview":["1","41","1681","68921","2825761","115856201","4750104241","194754273881"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["basis A013609"],"edges":["e-25d25e50279a57da"],"text":"A038324 \u2192 basis A013609"},"headline":"A038324 \u2192 basis A013609","chain":null}},{"id":"l-a52858692804fe87","kind":"fingerprint_related","node_a_id":"n-c5989f7729c933c8","node_b_id":"n-31f62c5b893224b0","fingerprint":"f6-640a8dc8403e6c62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-640a8dc8403e6c62"},"other_id":"n-31f62c5b893224b0","other":{"id":"n-31f62c5b893224b0","name":"reversal of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.","first_index":0,"fingerprint":"f6-640a8dc8403e6c62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","21"],["400","840","441"]],"card_rows":[["1"],["20","21"],["400","840","441"],["8000","25200","26460","9261"]],"row_sums_preview":["1","41","1681","68921","2825761","115856201","4750104241","194754273881"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d1088f7d1ff894","steps":["basis A013609","reverse"],"edges":["e-25d25e50279a57da","e-b2ce494c20cb57e7"],"text":"A038324 \u2192 basis A013609 \u2192 reverse"},"headline":"A038324 \u2192 basis A013609 \u2192 reverse","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:             0;\n  1:            20;\n  2:           840,            800;\n  3:         26460,          50400,           24000;\n  4:        740880,        2116800,         2016000,          640000;\n  5:      19448100,       74088000,       105840000,        67200000,        16000000;\n  6:     490092120,     2333772000,      4445280000,      4233600000,      2016000000,       384000000;\n  7:   12007256940,    68612896800,    163364040000,    207446400000,    148176000000,     56448000000,      8960000000;\n  8:  288174166560,  1921161110400,   5489031744000,   8712748800000,   8297856000000,   4741632000000,   1505280000000,   204800000000;\n  9: 6808114684980, 51871349980800, 172904499936000, 329341904640000, 392073696000000, 298722816000000, 142248960000000, 38707200000000, 4608000000000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (21*n+20*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 20.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 20;\nP[n_, x_] := P[n, x] = Expand[Together[((21*n + 20*n*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 20; for(n = 2, N, my(i = n + 1); v[i] = ((21*n + 20*n*x)*v[i - 1])/(n - 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-e1d1088f7d1ff894","steps":["basis A013609","D"],"edges":["e-25d25e50279a57da","e-cf836392720f7b39"],"text":"A038324 \u2192 basis A013609 \u2192 D"},"headline":"A038324 \u2192 basis A013609 \u2192 D","chain":null}