{"id":"n-160fd7a84ab6ab3d","revision":0,"first_index":1,"fingerprint":"f6-9c06be62c07bae00","name":"theta_plus_one_squared of Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["5","4"],["14","24"],["41","104","9"],["122","400","90"],["365","1452","567","16"],["1094","5096","2898","224"],["3281","17488","13158","1856","25"],["9842","59040","55404","11904","450"],["29525","196820","221490","65440","4625","36"],["88574","649528","852588","324544","35750","792"],["265721","2125752","3188727","1495616","231375","9720","49"],["797162","6908720","11658582","6524544","1325350","87984","1274"],["2391485","22320508","41851089","27280176","6944175","655956","18179","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","9","38","154","612","2400","9312","35808","136640","517936","1951776","7316960","27303616","101461632"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2304","-4512","-1552","176","0"],["1152","3216","232","-768","0"],["0","-480","688","748","-4"],["0","0","-208","-252","10"],["0","0","0","24","-8"],["0","0","0","0","2"]]},"terms":[{"coeff":{"coeffs":[["-5760","-12864","-11104","-3600","-176","0"],["2880","7824","5080","1196","20","0"],["0","-696","1420","1458","300","-2"],["0","0","-592","-674","-153","-3"],["0","0","0","48","0","0"],["0","0","0","0","5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["864","4752","5544","904","-840","-88","0"],["0","-864","-1272","-140","256","-12","0"],["0","0","-408","-740","-170","160","-2"],["0","0","0","72","60","-16","-4"],["0","0","0","0","6","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174727,"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-4c56b1482f79a238"]},{"id":174728,"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-4c56b1482f79a238"]},{"id":174729,"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-4c56b1482f79a238"]}],"region_exact":{},"incoming_edges":[{"id":"e-4c56b1482f79a238","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-160fd7a84ab6ab3d","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":"efa08e415d096c7933eb2d719f688a11f9a3d094cfe9980d4024fffe51656673","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-048e7c9e10af99dd","role":"sequence","name":"Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","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:    2;\n   3:    5,     4;\n   4:   14,    24;\n   5:   41,   104,     9;\n   6:  122,   400,    90;\n   7:  365,  1452,   567,    16;\n   8: 1094,  5096,  2898,   224;\n   9: 3281, 17488, 13158,  1856,  25;\n  10: 9842, 59040, 55404, 11904, 450;\n","recurrence_sentence":"The row polynomials satisfy (1152*n-2304-(480*n^2-3216*n+4512)*x-(208*n^3-688*n^2-232*n+1552)*x^2+(24*n^4-252*n^3+748*n^2-768*n+176)*x^3+(2*n^5-8*n^4+10*n^3-4*n^2)*x^4)*P(n,x) = (2880*n-5760-(696*n^2-7824*n+12864)*x-(592*n^3-1420*n^2-5080*n+11104)*x^2+(48*n^4-674*n^3+1458*n^2+1196*n-3600)*x^3+(5*n^5-153*n^3+300*n^2+20*n-176)*x^4+(n^5-3*n^3-2*n^2)*x^5)*P(n-1,x) + (864-(864*n-4752)*x-(408*n^2+1272*n-5544)*x^2+(72*n^3-740*n^2-140*n+904)*x^3+(6*n^4+60*n^3-170*n^2+256*n-840)*x^4+(4*n^4-16*n^3+160*n^2-12*n-88)*x^5-(2*n^4+4*n^3+2*n^2)*x^6)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2880*n - 5760 - (696*n^2 - 7824*n + 12864)*x - (592*n^3 - 1420*n^2 - 5080*n + 11104)*x^2 + (48*n^4 - 674*n^3 + 1458*n^2 + 1196*n - 3600)*x^3 + (5*n^5 - 153*n^3 + 300*n^2 + 20*n - 176)*x^4 + (n^5 - 3*n^3 - 2*n^2)*x^5)*P[n - 1, x] + (864 - (864*n - 4752)*x - (408*n^2 + 1272*n - 5544)*x^2 + (72*n^3 - 740*n^2 - 140*n + 904)*x^3 + (6*n^4 + 60*n^3 - 170*n^2 + 256*n - 840)*x^4 + (4*n^4 - 16*n^3 + 160*n^2 - 12*n - 88)*x^5 - (2*n^4 + 4*n^3 + 2*n^2)*x^6)*D[P[n - 1, x], x])/(1152*n - 2304 - (480*n^2 - 3216*n + 4512)*x - (208*n^3 - 688*n^2 - 232*n + 1552)*x^2 + (24*n^4 - 252*n^3 + 748*n^2 - 768*n + 176)*x^3 + (2*n^5 - 8*n^4 + 10*n^3 - 4*n^2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((2880*n - 5760 - (696*n^2 - 7824*n + 12864)*x - (592*n^3 - 1420*n^2 - 5080*n + 11104)*x^2 + (48*n^4 - 674*n^3 + 1458*n^2 + 1196*n - 3600)*x^3 + (5*n^5 - 153*n^3 + 300*n^2 + 20*n - 176)*x^4 + (n^5 - 3*n^3 - 2*n^2)*x^5)*v[i - 1] + (864 - (864*n - 4752)*x - (408*n^2 + 1272*n - 5544)*x^2 + (72*n^3 - 740*n^2 - 140*n + 904)*x^3 + (6*n^4 + 60*n^3 - 170*n^2 + 256*n - 840)*x^4 + (4*n^4 - 16*n^3 + 160*n^2 - 12*n - 88)*x^5 - (2*n^4 + 4*n^3 + 2*n^2)*x^6)*deriv(v[i - 1], x))/(1152*n - 2304 - (480*n^2 - 3216*n + 4512)*x - (208*n^3 - 688*n^2 - 232*n + 1552)*x^2 + (24*n^4 - 252*n^3 + 748*n^2 - 768*n + 176)*x^3 + (2*n^5 - 8*n^4 + 10*n^3 - 4*n^2)*x^4)); 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-048e7c9e10af99dd","steps":["(\u03b8+1)\u00b2"],"edges":["e-4c56b1482f79a238"],"text":"A201730 \u2192 (\u03b8+1)\u00b2"},"headline":"A201730 \u2192 (\u03b8+1)\u00b2","chain":null}