{"id":"n-f0c64936dd0c4060","revision":0,"first_index":0,"fingerprint":"f6-7d1144c6b7b8d949","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["144","480","300"],["1728","8640","10800","4000"],["20736","138240","259200","192000","50000"],["248832","2073600","5184000","5760000","3000000","600000"],["2985984","29859840","93312000","138240000","108000000","43200000","7000000"],["35831808","418037760","1567641600","2903040000","3024000000","1814400000","588000000","80000000"],["429981696","5733089280","25082265600","55738368000","72576000000","58060800000","28224000000","7680000000","900000000"],["5159780352","77396705280","386983526400","1003290624000","1567641600000","1567641600000","1016064000000","414720000000","97200000000","10000000000"],["61917364224","1031956070400","5804752896000","17199267840000","31352832000000","37623398400000","30481920000000","16588800000000","5832000000000","1200000000000","110000000000"],["743008370688","13621820129280","85136375808000","283787919360000","591224832000000","827714764800000","804722688000000","547430400000000","256608000000000","79200000000000","14520000000000","1200000000000"],["8916100448256","178322008965120","1225963811635200","4540606709760000","10642046976000000","17027275161600000","19313344512000000","15765995520000000","9237888000000000","3801600000000000","1045440000000000","172800000000000","13000000000000"],["106993205379072","2318186116546560","17386395874099200","70833464672256000","184462147584000000","332031865651200000","430411677696000000","409915883520000000","288222105600000000","148262400000000000","54362880000000000","13478400000000000","2028000000000000","140000000000000"],["1283918464548864","29958097506140160","243409542237388800","1081820187721728000","3098964079411200000","6197928158822400000","9038645231616000000","9837981204480000000","8070218956800000000","4981616640000000000","2283240960000000000","754790400000000000","170352000000000000","23520000000000000","1500000000000000"],["15407021574586368","385175539364659200","3370285969440768000","16227302815825920000","50710321299456000000","111562706858803200000","180772904632320000000","221354577100800000000","207519916032000000000","149448499200000000000","82196674560000000000","33965568000000000000","10221120000000000000","2116800000000000000","270000000000000000","16000000000000000"],["184884258895036416","4930246903867637760","46221064723759104000","239664780049121280000","811365140791296000000","1947276337899110400000","3470839768940544000000","4722230978150400000000","4980477984768000000000","4099158835200000000000","2630293585920000000000","1304277811200000000000","490613760000000000000","135475200000000000000","25920000000000000000","3072000000000000000","170000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["924","25168","660176","16866432","422597824","10430951168","254424504576","6146097836032","147286844570624","3505909808467968","82973198800408576","1953960399919480832","45815229377059405824","1070153259029585133568","24912172378804993458176"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","480","300"]],"recurrence":{"denominator":{"coeffs":[["6","10"],["0","5"]]},"terms":[{"coeff":{"coeffs":[["72","240","150"],["0","60","50"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193752,"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-251c1dc070a44d61"]},{"id":193753,"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-251c1dc070a44d61"]},{"id":193754,"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-251c1dc070a44d61"]},{"id":193755,"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-251c1dc070a44d61"]}],"region_exact":{},"incoming_edges":[{"id":"e-251c1dc070a44d61","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-f0c64936dd0c4060","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":"8efb348e48b0e79fbf6d0cd0e11d98ee3e8c4406b271c79edc9d1d7226504eb7","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-ec89fb3fc20684fc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(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-5399374268fff7f2","kind":"fingerprint_related","node_a_id":"n-67f904f032bfc830","node_b_id":"n-f0c64936dd0c4060","fingerprint":"f6-7d1144c6b7b8d949","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d1144c6b7b8d949"},"other_id":"n-67f904f032bfc830","other":{"id":"n-67f904f032bfc830","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-7d1144c6b7b8d949","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","10"],["36","120","75"]],"card_rows":[["1"],["6","10"],["36","120","75"],["216","1080","1350","500"]],"row_sums_preview":["1","16","231","3146","41261","527076","6603091","81491806"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["\u03b8+1"],"edges":["e-2225cd7ba75f4811"],"text":"A038259 \u2192 \u03b8+1"},"headline":"A038259 \u2192 \u03b8+1","chain":null}},{"id":"l-93a91733a8d96f4b","kind":"fingerprint_related","node_a_id":"n-4398082b97fd43de","node_b_id":"n-f0c64936dd0c4060","fingerprint":"f6-7d1144c6b7b8d949","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d1144c6b7b8d949"},"other_id":"n-4398082b97fd43de","other":{"id":"n-4398082b97fd43de","name":"theta_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-7d1144c6b7b8d949","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","120","75"],["216","1080","1350","500"],["1296","8640","16200","12000","3125"]],"card_rows":[["36","120","75"],["216","1080","1350","500"],["1296","8640","16200","12000","3125"],["7776","64800","162000","180000","93750","18750"]],"row_sums_preview":["231","3146","41261","527076","6603091","81491806","993845721","12004097336"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-7ae18e4d322020fd","e-ac4439b84583188d"],"text":"A013612 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A013612 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-e75dd6d88b88655e","kind":"fingerprint_related","node_a_id":"n-96e34c1114b4daa4","node_b_id":"n-f0c64936dd0c4060","fingerprint":"f6-7d1144c6b7b8d949","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d1144c6b7b8d949"},"other_id":"n-96e34c1114b4daa4","other":{"id":"n-96e34c1114b4daa4","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-7d1144c6b7b8d949","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","20"],["144","480","300"]],"card_rows":[["1"],["12","20"],["144","480","300"],["1728","8640","10800","4000"]],"row_sums_preview":["1","32","924","25168","660176","16866432","422597824","10430951168"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013609","\u03b8+1"],"edges":["e-e169cb33ba7b2c8c","e-d40b7b597c26b713"],"text":"A038271 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038271 \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:          144,            480,            300;\n  1:         1728,           8640,          10800,            4000;\n  2:        20736,         138240,         259200,          192000,           50000;\n  3:       248832,        2073600,        5184000,         5760000,         3000000,          600000;\n  4:      2985984,       29859840,       93312000,       138240000,       108000000,        43200000,         7000000;\n  5:     35831808,      418037760,     1567641600,      2903040000,      3024000000,      1814400000,       588000000,        80000000;\n  6:    429981696,     5733089280,    25082265600,     55738368000,     72576000000,     58060800000,     28224000000,      7680000000,       900000000;\n  7:   5159780352,    77396705280,   386983526400,   1003290624000,   1567641600000,   1567641600000,   1016064000000,    414720000000,     97200000000,    10000000000;\n  8:  61917364224,  1031956070400,  5804752896000,  17199267840000,  31352832000000,  37623398400000,  30481920000000,  16588800000000,   5832000000000,  1200000000000,   110000000000;\n  9: 743008370688, 13621820129280, 85136375808000, 283787919360000, 591224832000000, 827714764800000, 804722688000000, 547430400000000, 256608000000000, 79200000000000, 14520000000000, 1200000000000;\n","recurrence_sentence":"The row polynomials satisfy (6+(5*n+10)*x)*P(n,x) = (72+(60*n+240)*x+(50*n+150)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 480*x + 300*x^2.","mathematica":"P[0, x_] := 144 + 480*x + 300*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((72 + (60*n + 240)*x + (50*n + 150)*x^2)*P[n - 1, x])/(6 + (5*n + 10)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 480*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (60*n + 240)*x + (50*n + 150)*x^2)*v[i - 1])/(6 + (5*n + 10)*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-ec89fb3fc20684fc","steps":["\u03b8+1"],"edges":["e-251c1dc070a44d61"],"text":"A038336 \u2192 \u03b8+1"},"headline":"A038336 \u2192 \u03b8+1","chain":null}