{"id":"n-9212bcd5fb1e697f","revision":0,"first_index":1,"fingerprint":"f6-48c328a3a13ffc6f","name":"derivative of Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.","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":false,"rows":[[],["3"],["15","30"],["63","420","315"],["255","4410","9450","3780"],["1023","42240","197505","207900","51975"],["4095","390390","3603600","7567560","4729725","810810"],["16383","3554460","61756695","232432200","274324050","113513400","14189175"],["65535","32153970","1025428950","6551885340","13071608550","9841611780","2894591700","275675400"],["262143","290041080","16744479885","176102280900","560666331225","686156070600","357482074950","78567489000","5892561675"],["1048575","2612991150","270957109500","4603490781120","22605394286475","42368042226510","34886584032300","13309332636600","2268636244875","137493105750"],["4194303","23527406100","4362750159075","118338754842000","877092192553500","2428678219968000","2974182117031125","1755852433335000","511350609594825","69571511509500","3478575575475"],["16777215","211788597930","70051040309250","3010821872958900","33202476811003500","132687870982266600","232849524339432000","200008156802454000","88666406722018125","20349667116528750","2261074124058750","94870242967500"],["67108863","1906265153520","1123040425226265","76111159307683500","1236687965949310875","7019664316382718000","17224407299913313500","20724594316736310000","13141815023854266750","4531192544613735000","840441251912637375","77698728990382500","2774954606799375"],["268435455","17157057470310","17988662587732200","1916255467794802680","45567295214655839625","363255883771562431650","1225206192731149629000","2013337767196814058000","1757169705905665301250","854916648454861492500","235662711654147277500","36052210251537480000","2816578925901365625","86663966950811250"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","45","798","17895","500643","17106180","699786363","33693021225","1881901591458","120452544462855","8742930696168903","713201131859372520","64876512509024645523","6536005919360954127045"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","-10","-15"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-5","-14","-6"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-3"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","10"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189703,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-8cb6ffd92e488afb"]},{"id":189704,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-8cb6ffd92e488afb"]},{"id":189705,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-8cb6ffd92e488afb"]}],"region_exact":{},"incoming_edges":[{"id":"e-8cb6ffd92e488afb","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-9212bcd5fb1e697f","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":"1f6682047c4ea2ad95227024a232a9f2ac7817c0620329bc60234cfb60947c52","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-67b93749ab954f15","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      0;\n   2:      3;\n   3:     15,        30;\n   4:     63,       420,         315;\n   5:    255,      4410,        9450,         3780;\n   6:   1023,     42240,      197505,       207900,        51975;\n   7:   4095,    390390,     3603600,      7567560,      4729725,       810810;\n   8:  16383,   3554460,    61756695,    232432200,    274324050,    113513400,     14189175;\n   9:  65535,  32153970,  1025428950,   6551885340,  13071608550,   9841611780,   2894591700,   275675400;\n  10: 262143, 290041080, 16744479885, 176102280900, 560666331225, 686156070600, 357482074950, 78567489000, 5892561675;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4+10*x+15*x^2)*P(n-2,x) - (5*x+14*x^2+6*x^3)*d/dx P(n-2,x) - (x^2+3*x^3)*d^2/dx^2 P(n-2,x) + (5+10*x)*P(n-1,x) + (5*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(4 + 10*x + 15*x^2)*P[n - 2, x] - (5*x + 14*x^2 + 6*x^3)*D[P[n - 2, x], x] - (x^2 + 3*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 10*x)*P[n - 1, x] + (5*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = -(4 + 10*x + 15*x^2)*v[i - 2] - (5*x + 14*x^2 + 6*x^3)*deriv(v[i - 2], x) - (x^2 + 3*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 10*x)*v[i - 1] + (5*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-67b93749ab954f15","steps":["D"],"edges":["e-8cb6ffd92e488afb"],"text":"A156289 \u2192 D"},"headline":"A156289 \u2192 D","chain":null}