{"id":"n-e3e850d58b0f9c5f","revision":0,"first_index":0,"fingerprint":"f6-4df2d4bb71a9e1dc","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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","528","363"],["1728","9504","13068","5324"],["20736","152064","313632","255552","73205"],["248832","2280960","6272640","7666560","4392300","966306"],["2985984","32845824","112907520","183997440","158122800","69574032","12400927"],["35831808","459841536","1896846336","3863946240","4427438400","2922109344","1041677868","155897368"],["429981696","6306398208","30349541376","74187767808","106258521600","93507499008","50000537664","14966147328","1929229929"],["5159780352","85136375808","468250066944","1335379820544","2295184066560","2524702473216","1800019355904","808171955712","208356832332","23579476910"],["61917364224","1135151677440","7023751004160","22892225495040","45903681331200","60592859357184","54000580677120","32326878228480","12501409939920","2829537229200","285311670611"],["743008370688","14984002142208","103015014727680","377721720668160","865612276531200","1333042905858048","1425615329875968","1066786981539840","550062037356480","186749457127200","37661140520652","3423740047332"],["8916100448256","196154209861632","1483416212078592","6043547530690560","15581020977561600","27422596920508416","34214767917023232","30723465068347392","19802233344833280","8963973942105600","2711602117486944","493018566815808","40799568897373"],["106993205379072","2550004728201216","21037539007660032","94279341478772736","270071030277734400","534740639949914112","762500542150803456","798810091777032192","617829680358798336","349594983742118400","141003310109321088","38455448211633024","6364732747990188","483317970015034"],["1283918464548864","32953907256754176","294525546107240448","1439902669857619968","4537193308665937920","9981825279065063424","16012511385166872576","19171442202648772608","17299231050046353408","11746391453735178240","5922139024591485696","2153505099851449344","534637550831175792","81197418962525712","5696247503748615"],["15407021574586368","423693093301125120","4078046023023329280","21598540047864299520","74244981414533529600","179672855023171141632","320250227703337451520","431357449559597383680","444837369858334801920","352391743612055347200","213197004885293485056","96907729493315220480","32078253049870547520","7307767706627314080","1025324550674750700","66835970710650416"],["184884258895036416","5423271594254401536","55927488315748515840","318993822245380423680","1187919702632536473600","3136108014949896290304","6148804371904079069184","9202292257271410851840","10676096876600035246080","9665602110502089523200","6822304156329391521792","3721256812543304466432","1539756146393786280960","467697133224148101120","98431156864776067200","12832506376444879872","781145407680726737"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1035","29624","815189","21827598","572834527","14803588900","377935624617","9553940204282","239553303974579","5965417614765456","147685512476658685","3637827655715373286","89214436062754726791","2179453229013284964092","53060411857413857875793"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","528","363"]],"recurrence":{"denominator":{"coeffs":[["12","22"],["0","11"]]},"terms":[{"coeff":{"coeffs":[["144","528","363"],["0","132","121"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193771,"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-9f686dc9d673d92c"]},{"id":193772,"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-9f686dc9d673d92c"]},{"id":193773,"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-9f686dc9d673d92c"]},{"id":193774,"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-9f686dc9d673d92c"]}],"region_exact":{},"incoming_edges":[{"id":"e-9f686dc9d673d92c","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-e3e850d58b0f9c5f","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":"e811730f88ef81474069f35c48d3199a8a3272748ce9ef5d72f4193cefb78df7","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-344e4bfa9933088a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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":[],"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,            528,             363;\n  1:         1728,           9504,           13068,            5324;\n  2:        20736,         152064,          313632,          255552,           73205;\n  3:       248832,        2280960,         6272640,         7666560,         4392300,           966306;\n  4:      2985984,       32845824,       112907520,       183997440,       158122800,         69574032,         12400927;\n  5:     35831808,      459841536,      1896846336,      3863946240,      4427438400,       2922109344,       1041677868,        155897368;\n  6:    429981696,     6306398208,     30349541376,     74187767808,    106258521600,      93507499008,      50000537664,      14966147328,      1929229929;\n  7:   5159780352,    85136375808,    468250066944,   1335379820544,   2295184066560,    2524702473216,    1800019355904,     808171955712,    208356832332,     23579476910;\n  8:  61917364224,  1135151677440,   7023751004160,  22892225495040,  45903681331200,   60592859357184,   54000580677120,   32326878228480,  12501409939920,   2829537229200,   285311670611;\n  9: 743008370688, 14984002142208, 103015014727680, 377721720668160, 865612276531200, 1333042905858048, 1425615329875968, 1066786981539840, 550062037356480, 186749457127200, 37661140520652, 3423740047332;\n","recurrence_sentence":"The row polynomials satisfy (12+(11*n+22)*x)*P(n,x) = (144+(132*n+528)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 528*x + 363*x^2.","mathematica":"P[0, x_] := 144 + 528*x + 363*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((144 + (132*n + 528)*x + (121*n + 363)*x^2)*P[n - 1, x])/(12 + (11*n + 22)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 528*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144 + (132*n + 528)*x + (121*n + 363)*x^2)*v[i - 1])/(12 + (11*n + 22)*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-344e4bfa9933088a","steps":["\u03b8+1"],"edges":["e-9f686dc9d673d92c"],"text":"A038337 \u2192 \u03b8+1"},"headline":"A038337 \u2192 \u03b8+1","chain":null}