{"id":"n-6fe6856f0b35d23f","revision":0,"first_index":1,"fingerprint":"f6-135fd721cf11d97a","name":"one_plus_derivative of theta_plus_one 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"],["3","2"],["9","14","3"],["19","78","42","4"],["33","284","366","100","5"],["51","770","2040","1260","205","6"],["73","1722","8025","9960","3570","378","7"],["99","3374","24738","53900","38010","8820","644","8"],["129","6008","64092","220584","271810","121968","19656","1032","9"],["163","9954","146016","735000","1433250","1117116","343728","40392","1575","10"],["201","15590","301455","2097120","6003480","7407036","3933930","875160","77715","2310","11"],["243","23342","575850","5304420","21071160","38264688","32144112","12269400","2052765","141570","3278","12"],["289","33684","1033098","12189100","64358415","162698976","201502224","121472208","34684650","4498780","246246","4524","13"],["339","47138","1759992","25904164","175790615","592594002","1024287264","911216592","410143734","90384580","9309872","411684","6097","14"],["393","64274","2871141","51597000","438127690","1903956054","4400292897","5473950768","3637383750","1260959700","219893674","18341232","665028","8050","15"]],"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:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["54","-142","70","-8"],["-300","684","-304","32"],["650","-1198","446","-40"],["-712","960","-260","16"],["420","-352","48","0"],["-128","48","0","0"],["16","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["18","690","-2666","1604","-208"],["-108","-3668","10440","-5304","608"],["262","7558","-15114","5876","-528"],["-336","-7868","10396","-2684","144"],["244","4400","-3432","432","0"],["-96","-1256","432","0","0"],["16","144","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-78","890","-2478","1706","-248"],["0","374","-3834","8264","-4434","536"],["0","-660","6052","-9080","3044","-208"],["0","548","-4496","4048","-624","0"],["0","-216","1584","-624","0","0"],["0","32","-208","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-21","167","-401","288","-48"],["0","0","91","-615","1155","-624","80"],["0","0","-136","766","-948","240","0"],["0","0","86","-404","240","0","0"],["0","0","-20","80","0","0","0"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176528,"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-7a49159f9053f379"]},{"id":176529,"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-7a49159f9053f379"]},{"id":176530,"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-7a49159f9053f379"]}],"region_exact":{},"incoming_edges":[{"id":"e-7a49159f9053f379","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-6fe6856f0b35d23f","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":"7e2f1695d6e0fee660ad4c2c40d1a16ec7de171f2f6fa7a03cff16de4b00b897","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-012ae01d9db4fd27","role":"sequence","name":"theta_plus_one of 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:   3,    2;\n   3:   9,   14,      3;\n   4:  19,   78,     42,      4;\n   5:  33,  284,    366,    100,       5;\n   6:  51,  770,   2040,   1260,     205,       6;\n   7:  73, 1722,   8025,   9960,    3570,     378,      7;\n   8:  99, 3374,  24738,  53900,   38010,    8820,    644,     8;\n   9: 129, 6008,  64092, 220584,  271810,  121968,  19656,  1032,    9;\n  10: 163, 9954, 146016, 735000, 1433250, 1117116, 343728, 40392, 1575, 10;\n","recurrence_sentence":"The row polynomials satisfy (16*n^6-128*n^5+420*n^4-712*n^3+650*n^2-300*n+54+(48*n^5-352*n^4+960*n^3-1198*n^2+684*n-142)*x+(48*n^4-260*n^3+446*n^2-304*n+70)*x^2+(16*n^3-40*n^2+32*n-8)*x^3)*P(n,x) = (16*n^6-96*n^5+244*n^4-336*n^3+262*n^2-108*n+18+(144*n^6-1256*n^5+4400*n^4-7868*n^3+7558*n^2-3668*n+690)*x+(432*n^5-3432*n^4+10396*n^3-15114*n^2+10440*n-2666)*x^2+(432*n^4-2684*n^3+5876*n^2-5304*n+1604)*x^3+(144*n^3-528*n^2+608*n-208)*x^4)*P(n-1,x) + ((32*n^5-216*n^4+548*n^3-660*n^2+374*n-78)*x-(208*n^5-1584*n^4+4496*n^3-6052*n^2+3834*n-890)*x^2-(624*n^4-4048*n^3+9080*n^2-8264*n+2478)*x^3-(624*n^3-3044*n^2+4434*n-1706)*x^4-(208*n^2-536*n+248)*x^5)*d/dx P(n-1,x) - ((20*n^4-86*n^3+136*n^2-91*n+21)*x^2-(80*n^4-404*n^3+766*n^2-615*n+167)*x^3-(240*n^3-948*n^2+1155*n-401)*x^4-(240*n^2-624*n+288)*x^5-(80*n-48)*x^6)*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[((16*n^6 - 96*n^5 + 244*n^4 - 336*n^3 + 262*n^2 - 108*n + 18 + (144*n^6 - 1256*n^5 + 4400*n^4 - 7868*n^3 + 7558*n^2 - 3668*n + 690)*x + (432*n^5 - 3432*n^4 + 10396*n^3 - 15114*n^2 + 10440*n - 2666)*x^2 + (432*n^4 - 2684*n^3 + 5876*n^2 - 5304*n + 1604)*x^3 + (144*n^3 - 528*n^2 + 608*n - 208)*x^4)*P[n - 1, x] + ((32*n^5 - 216*n^4 + 548*n^3 - 660*n^2 + 374*n - 78)*x - (208*n^5 - 1584*n^4 + 4496*n^3 - 6052*n^2 + 3834*n - 890)*x^2 - (624*n^4 - 4048*n^3 + 9080*n^2 - 8264*n + 2478)*x^3 - (624*n^3 - 3044*n^2 + 4434*n - 1706)*x^4 - (208*n^2 - 536*n + 248)*x^5)*D[P[n - 1, x], x] - ((20*n^4 - 86*n^3 + 136*n^2 - 91*n + 21)*x^2 - (80*n^4 - 404*n^3 + 766*n^2 - 615*n + 167)*x^3 - (240*n^3 - 948*n^2 + 1155*n - 401)*x^4 - (240*n^2 - 624*n + 288)*x^5 - (80*n - 48)*x^6)*D[P[n - 1, x], {x, 2}])/(16*n^6 - 128*n^5 + 420*n^4 - 712*n^3 + 650*n^2 - 300*n + 54 + (48*n^5 - 352*n^4 + 960*n^3 - 1198*n^2 + 684*n - 142)*x + (48*n^4 - 260*n^3 + 446*n^2 - 304*n + 70)*x^2 + (16*n^3 - 40*n^2 + 32*n - 8)*x^3)]];\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] = ((16*n^6 - 96*n^5 + 244*n^4 - 336*n^3 + 262*n^2 - 108*n + 18 + (144*n^6 - 1256*n^5 + 4400*n^4 - 7868*n^3 + 7558*n^2 - 3668*n + 690)*x + (432*n^5 - 3432*n^4 + 10396*n^3 - 15114*n^2 + 10440*n - 2666)*x^2 + (432*n^4 - 2684*n^3 + 5876*n^2 - 5304*n + 1604)*x^3 + (144*n^3 - 528*n^2 + 608*n - 208)*x^4)*v[i - 1] + ((32*n^5 - 216*n^4 + 548*n^3 - 660*n^2 + 374*n - 78)*x - (208*n^5 - 1584*n^4 + 4496*n^3 - 6052*n^2 + 3834*n - 890)*x^2 - (624*n^4 - 4048*n^3 + 9080*n^2 - 8264*n + 2478)*x^3 - (624*n^3 - 3044*n^2 + 4434*n - 1706)*x^4 - (208*n^2 - 536*n + 248)*x^5)*deriv(v[i - 1], x) - ((20*n^4 - 86*n^3 + 136*n^2 - 91*n + 21)*x^2 - (80*n^4 - 404*n^3 + 766*n^2 - 615*n + 167)*x^3 - (240*n^3 - 948*n^2 + 1155*n - 401)*x^4 - (240*n^2 - 624*n + 288)*x^5 - (80*n - 48)*x^6)*deriv(deriv(v[i - 1], x), x))/(16*n^6 - 128*n^5 + 420*n^4 - 712*n^3 + 650*n^2 - 300*n + 54 + (48*n^5 - 352*n^4 + 960*n^3 - 1198*n^2 + 684*n - 142)*x + (48*n^4 - 260*n^3 + 446*n^2 - 304*n + 70)*x^2 + (16*n^3 - 40*n^2 + 32*n - 8)*x^3)); 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-5c0c5bce3aee8449","steps":["\u03b8+1","D+1"],"edges":["e-79b5bc170ae8a162","e-7a49159f9053f379"],"text":"A391045 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A391045 \u2192 \u03b8+1 \u2192 D+1","chain":null}