{"id":"n-3b22cec39202b5b9","revision":0,"first_index":1,"fingerprint":"f6-e5341cf5487df687","name":"one_plus_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":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","3"],["16","45","15"],["64","483","525","105"],["256","4665","11655","6930","945"],["1024","43263","218625","273735","103950","10395"],["4096","394485","3798795","8768760","6621615","1756755","135135"],["16384","3570843","63533925","253017765","332432100","168378210","33108075","2027025"],["65536","32219505","1041505935","6893694990","14709579885","12455933490","4534860330","689188500","34459425"],["262144","290303223","16889500425","181683774195","604691901450","798289336845","471841420050","129636356850","15713497800","654729075"],["1048576","2614039725","272263605075","4693809817620","23756266981755","46889121083805","41947924403385","18293130355500","3932302824450","389563799625","13749310575"],["4194304","23531600403","4374513862125","119793004895025","906676881264000","2604096658478700","3378961820359125","2180735592910875","730832163761700","126388245908925","10435726726425","316234143225"],["16777216","211805375145","70156934608215","3034172219728650","33955182279243225","139328366344467300","254964169503143100","233272374565230000","113667426322324875","30201490085641875","4296040835711625","300422436063750","7905853580625"],["67108864","1906332262383","1123993557803025","76485506116092255","1255715755776231750","7267001909572580175","18394351352643766500","23185223931009640500","15732389313446305500","5991394213930875750","1293560506374010875","154102479164258625","9249848689331250","213458046676875"],["268435456","17157325905765","17997241116467355","1922251688657380080","46046359081604540295","372369342814493599575","1285748840026410034275","2188367223301264005000","2008836926805267058500","1050157726888824303750","321154376499633426750","57476093129187232500","5820929780196155625","303323884327839375","6190283353629375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","76","1177","24451","650992","21479641","856084327","40391507596","2219691082057","140190747270091","10062634378104832","813097919201895601","73360814174726944327","7338227598581930013676"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-4","-5","-2","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-5","-2","3","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","0","1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","5","-1"]]},"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":189706,"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-46e422e5b2b45b39"]},{"id":189707,"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-46e422e5b2b45b39"]},{"id":189708,"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-46e422e5b2b45b39"]}],"region_exact":{},"incoming_edges":[{"id":"e-46e422e5b2b45b39","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-3b22cec39202b5b9","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":"eb69f14ea9ee7a2485ab133f5732e07bd5f6ec30bcc2bbf99c3f5e1ebf826df5","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:      1;\n   2:      4,         3;\n   3:     16,        45,          15;\n   4:     64,       483,         525,          105;\n   5:    256,      4665,       11655,         6930,          945;\n   6:   1024,     43263,      218625,       273735,       103950,        10395;\n   7:   4096,    394485,     3798795,      8768760,      6621615,      1756755,       135135;\n   8:  16384,   3570843,    63533925,    253017765,    332432100,    168378210,     33108075,      2027025;\n   9:  65536,  32219505,  1041505935,   6893694990,  14709579885,  12455933490,   4534860330,    689188500,    34459425;\n  10: 262144, 290303223, 16889500425, 181683774195, 604691901450, 798289336845, 471841420050, 129636356850, 15713497800, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4+5*x+2*x^2-3*x^3)*P(n-2,x) - (5*x+2*x^2-3*x^3-2*x^4)*d/dx P(n-2,x) - (x^2-x^4)*d^2/dx^2 P(n-2,x) + (5+5*x-x^2)*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) = 1, P(2,x) = 4 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(4 + 5*x + 2*x^2 - 3*x^3)*P[n - 2, x] - (5*x + 2*x^2 - 3*x^3 - 2*x^4)*D[P[n - 2, x], x] - (x^2 - x^4)*D[P[n - 2, x], {x, 2}] + (5 + 5*x - x^2)*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] = 1; v[2] = 4 + 3*x; for(n = 3, N, my(i = n); v[i] = -(4 + 5*x + 2*x^2 - 3*x^3)*v[i - 2] - (5*x + 2*x^2 - 3*x^3 - 2*x^4)*deriv(v[i - 2], x) - (x^2 - x^4)*deriv(deriv(v[i - 2], x), x) + (5 + 5*x - x^2)*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+1"],"edges":["e-46e422e5b2b45b39"],"text":"A156289 \u2192 D+1"},"headline":"A156289 \u2192 D+1","chain":null}