{"id":"n-634cee2f3de2454f","revision":1,"first_index":1,"fingerprint":"f6-4635e8cd337c1e83","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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"],["1","4"],["1","16","9"],["1","36","90","16"],["1","64","378","320","25"],["1","100","1080","2240","875","36"],["1","144","2475","9600","9450","2016","49"],["1","196","4914","30800","57750","31752","4116","64"],["1","256","8820","81536","250250","266112","90552","7680","81"],["1","324","14688","188160","859950","1513512","1009008","228096","13365","100"],["1","400","23085","391680","2499000","6604416","7357350","3294720","521235","22000","121"],["1","484","34650","752400","6395400","23750496","40023984","30201600","9555975","1101100","34606","144"],["1","576","50094","1355200","14812875","73675008","175489776","201636864","108301050","25168000","2180178","52416","169"],["1","676","70200","2315456","31656625","203351148","651819168","1064194560","875466306","347633000","61183408","4088448","76895","196"],["1","784","95823","3785600","63313250","510558048","2122006887","4682456064","5499724230","3362559200","1017174158","139007232","7320404","109760","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","26","143","788","4332","23735","129593","705288","3827204","20714008","111850840","602722207","3241856087","17408111666"],"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"]],"recurrence":{"denominator":{"coeffs":[["-2","4"],["12","-32"],["-26","104"],["24","-176"],["-8","164"],["0","-80"],["0","16"]]},"terms":[{"coeff":{"coeffs":[["-2","-100","-128"],["12","480","512"],["-26","-768","-548"],["24","424","-316"],["-8","20","1000"],["0","-80","-664"],["0","16","144"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-62","-160"],["0","-34","258","544"],["0","44","-324","-380"],["0","-16","148","-500"],["0","0","-80","704"],["0","0","32","-208"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","3","-4","-32"],["0","0","-11","20","96"],["0","0","10","-30","-40"],["0","0","0","26","-104"],["0","0","0","-20","80"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164594,"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-e9c68bdfe2122a78"]},{"id":164595,"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-e9c68bdfe2122a78"]},{"id":164596,"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-e9c68bdfe2122a78"]}],"region_exact":{},"incoming_edges":[{"id":"e-e9c68bdfe2122a78","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-634cee2f3de2454f","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":"36f4b3ea4f4126df4328ea36a11e3e437a2950bb15b390425935c8a9c1dbf202","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-5c0c5bce3aee8449","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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: 1,   4;\n   3: 1,  16,     9;\n   4: 1,  36,    90,     16;\n   5: 1,  64,   378,    320,     25;\n   6: 1, 100,  1080,   2240,    875,      36;\n   7: 1, 144,  2475,   9600,   9450,    2016,      49;\n   8: 1, 196,  4914,  30800,  57750,   31752,    4116,     64;\n   9: 1, 256,  8820,  81536, 250250,  266112,   90552,   7680,    81;\n  10: 1, 324, 14688, 188160, 859950, 1513512, 1009008, 228096, 13365, 100;\n","recurrence_sentence":"The row polynomials satisfy -(8*n^4-24*n^3+26*n^2-12*n+2-(16*n^6-80*n^5+164*n^4-176*n^3+104*n^2-32*n+4)*x)*P(n,x) = -(8*n^4-24*n^3+26*n^2-12*n+2-(16*n^6-80*n^5+20*n^4+424*n^3-768*n^2+480*n-100)*x-(144*n^6-664*n^5+1000*n^4-316*n^3-548*n^2+512*n-128)*x^2)*P(n-1,x) - ((16*n^3-44*n^2+34*n-8)*x-(32*n^5-80*n^4+148*n^3-324*n^2+258*n-62)*x^2+(208*n^5-704*n^4+500*n^3+380*n^2-544*n+160)*x^3)*d/dx P(n-1,x) + ((10*n^2-11*n+3)*x^2-(20*n^4-26*n^3+30*n^2-20*n+4)*x^3+(80*n^4-104*n^3-40*n^2+96*n-32)*x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^4 - 24*n^3 + 26*n^2 - 12*n + 2 - (16*n^6 - 80*n^5 + 20*n^4 + 424*n^3 - 768*n^2 + 480*n - 100)*x - (144*n^6 - 664*n^5 + 1000*n^4 - 316*n^3 - 548*n^2 + 512*n - 128)*x^2)*P[n - 1, x] - ((16*n^3 - 44*n^2 + 34*n - 8)*x - (32*n^5 - 80*n^4 + 148*n^3 - 324*n^2 + 258*n - 62)*x^2 + (208*n^5 - 704*n^4 + 500*n^3 + 380*n^2 - 544*n + 160)*x^3)*D[P[n - 1, x], x] + ((10*n^2 - 11*n + 3)*x^2 - (20*n^4 - 26*n^3 + 30*n^2 - 20*n + 4)*x^3 + (80*n^4 - 104*n^3 - 40*n^2 + 96*n - 32)*x^4)*D[P[n - 1, x], {x, 2}])/(-8*n^4 + 24*n^3 - 26*n^2 + 12*n - 2 + (16*n^6 - 80*n^5 + 164*n^4 - 176*n^3 + 104*n^2 - 32*n + 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(8*n^4 - 24*n^3 + 26*n^2 - 12*n + 2 - (16*n^6 - 80*n^5 + 20*n^4 + 424*n^3 - 768*n^2 + 480*n - 100)*x - (144*n^6 - 664*n^5 + 1000*n^4 - 316*n^3 - 548*n^2 + 512*n - 128)*x^2)*v[i - 1] - ((16*n^3 - 44*n^2 + 34*n - 8)*x - (32*n^5 - 80*n^4 + 148*n^3 - 324*n^2 + 258*n - 62)*x^2 + (208*n^5 - 704*n^4 + 500*n^3 + 380*n^2 - 544*n + 160)*x^3)*deriv(v[i - 1], x) + ((10*n^2 - 11*n + 3)*x^2 - (20*n^4 - 26*n^3 + 30*n^2 - 20*n + 4)*x^3 + (80*n^4 - 104*n^3 - 40*n^2 + 96*n - 32)*x^4)*deriv(deriv(v[i - 1], x), x))/(-8*n^4 + 24*n^3 - 26*n^2 + 12*n - 2 + (16*n^6 - 80*n^5 + 164*n^4 - 176*n^3 + 104*n^2 - 32*n + 4)*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-79ebafb266da3b41","steps":["\u2299 C(n,k+1) (shift -1)","(\u03b8+1)\u00b2"],"edges":["e-1491609a6fc5f8c9","e-e9c68bdfe2122a78"],"text":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 (\u03b8+1)\u00b2"},"headline":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 (\u03b8+1)\u00b2","chain":null}