{"id":"n-b82a18a2f7e9534b","revision":0,"first_index":0,"fingerprint":"f6-65a6331fbe5d08da","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","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"],["10","8"],["100","160","48"],["1000","2400","1440","256"],["10000","32000","28800","10240","1280"],["100000","400000","480000","256000","64000","6144"],["1000000","4800000","7200000","5120000","1920000","368640","28672"],["10000000","56000000","100800000","89600000","44800000","12902400","2007040","131072"],["100000000","640000000","1344000000","1433600000","896000000","344064000","80281600","10485760","589824"],["1000000000","7200000000","17280000000","21504000000","16128000000","7741440000","2408448000","471859200","53084160","2621440"],["10000000000","80000000000","216000000000","307200000000","268800000000","154828800000","60211200000","15728640000","2654208000","262144000","11534336"],["100000000000","880000000000","2640000000000","4224000000000","4224000000000","2838528000000","1324646400000","432537600000","97320960000","14417920000","1268776960","50331648"],["1000000000000","9600000000000","31680000000000","56320000000000","63360000000000","48660480000000","26492928000000","10380902400000","2919628800000","576716800000","76126617600","6039797760","218103808"],["10000000000000","104000000000000","374400000000000","732160000000000","915200000000000","790732800000000","492011520000000","224919552000000","75910348800000","18743296000000","3298820096000","392586854400","28353495040","939524096"],["100000000000000","1120000000000000","4368000000000000","9318400000000000","12812800000000000","12300288000000000","8610201600000000","4498391040000000","1771241472000000","524812288000000","115458703360000","18320719872000","1984744652800","131533373440","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","308","5096","82320","1306144","20437312","316240512","4849021184","73789452800","1115696526336","16776769988608","251073040519168","3741798216769536","55560034127790080"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["5","0"],["0","2"]]},"terms":[{"coeff":{"coeffs":[["50","40","8"],["0","20","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188070,"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-29d5863d60295c31"]},{"id":188071,"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-29d5863d60295c31"]},{"id":188072,"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-29d5863d60295c31"]},{"id":188073,"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-29d5863d60295c31"]}],"region_exact":{},"incoming_edges":[{"id":"e-29d5863d60295c31","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-b82a18a2f7e9534b","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"928fe047a434ccefd1e385161fd52060813ab6c9f9bd16f8950098d09ca6fefe","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-8afa7b2308002172","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^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-26759cd0fc5a2326","kind":"fingerprint_related","node_a_id":"n-b82a18a2f7e9534b","node_b_id":"n-1fd1b3bc25cf1bde","fingerprint":"f6-65a6331fbe5d08da","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-65a6331fbe5d08da"},"other_id":"n-1fd1b3bc25cf1bde","other":{"id":"n-1fd1b3bc25cf1bde","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-65a6331fbe5d08da","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","640","192"],["8000","19200","11520","2048"],["160000","512000","460800","163840","20480"]],"card_rows":[["400","640","192"],["8000","19200","11520","2048"],["160000","512000","460800","163840","20480"],["3200000","12800000","15360000","8192000","2048000","196608"]],"row_sums_preview":["1232","40768","1317120","41796608","1307987968","40478785536","1241349423104","37780199833600"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-4a3b4bf6a80699a3","e-b300226b4c277954"],"text":"A038334 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038334 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-4a755090f7fd90ff","kind":"fingerprint_related","node_a_id":"n-0b5b32088ca1fd1c","node_b_id":"n-b82a18a2f7e9534b","fingerprint":"f6-65a6331fbe5d08da","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-65a6331fbe5d08da"},"other_id":"n-0b5b32088ca1fd1c","other":{"id":"n-0b5b32088ca1fd1c","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.","first_index":0,"fingerprint":"f6-65a6331fbe5d08da","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","4"],["25","40","12"]],"card_rows":[["1"],["5","4"],["25","40","12"],["125","300","180","32"]],"row_sums_preview":["1","9","77","637","5145","40817","319333","2470629"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A007318","\u03b8+1"],"edges":["e-7f6faf86ff30c5bb","e-4a6585cd7a37314e"],"text":"A038220 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038220 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-d9efbb24c8832a03","kind":"fingerprint_related","node_a_id":"n-b82a18a2f7e9534b","node_b_id":"n-d8e5c4e43f9340d6","fingerprint":"f6-65a6331fbe5d08da","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-65a6331fbe5d08da"},"other_id":"n-d8e5c4e43f9340d6","other":{"id":"n-d8e5c4e43f9340d6","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-65a6331fbe5d08da","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","360","108"],["3375","8100","4860","864"],["50625","162000","145800","51840","6480"]],"card_rows":[["225","360","108"],["3375","8100","4860","864"],["50625","162000","145800","51840","6480"],["759375","3037500","3645000","1944000","486000","46656"]],"row_sums_preview":["693","17199","416745","9918531","232793757","5403265623","124275109329","2836714452075"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013609","\u03b8+1"],"edges":["e-f3e6c91e6b1bdc2d","e-df802384c5e9632f"],"text":"A038329 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038329 \u2192 basis A013609 \u2192 \u03b8+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:         10,          8;\n  2:        100,        160,          48;\n  3:       1000,       2400,        1440,         256;\n  4:      10000,      32000,       28800,       10240,        1280;\n  5:     100000,     400000,      480000,      256000,       64000,       6144;\n  6:    1000000,    4800000,     7200000,     5120000,     1920000,     368640,      28672;\n  7:   10000000,   56000000,   100800000,    89600000,    44800000,   12902400,    2007040,    131072;\n  8:  100000000,  640000000,  1344000000,  1433600000,   896000000,  344064000,   80281600,  10485760,   589824;\n  9: 1000000000, 7200000000, 17280000000, 21504000000, 16128000000, 7741440000, 2408448000, 471859200, 53084160, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (5+2*n*x)*P(n,x) = (50+(20*n+40)*x+(8*n+8)*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[((50 + (20*n + 40)*x + (8*n + 8)*x^2)*P[n - 1, x])/(5 + 2*n*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] = ((50 + (20*n + 40)*x + (8*n + 8)*x^2)*v[i - 1])/(5 + 2*n*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-3dd184aba99388f7","steps":["basis A013611","\u03b8+1"],"edges":["e-8c5f302cdd82df95","e-29d5863d60295c31"],"text":"A038291 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038291 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}