{"id":"n-32c953bc446d5813","revision":0,"first_index":0,"fingerprint":"f6-e58e8e6744652fd9","name":"one_plus_derivative of Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["8","4"],["36","40","8"],["192","336","144","16"],["1200","2880","1920","448","32"],["8640","26400","24000","8640","1280","64"],["70560","262080","302400","147840","33600","3456","128"],["645120","2822400","3951360","2446080","752640","118272","8960","256"],["6531840","32901120","54190080","40642560","15805440","3354624","387072","22528","512"],["72576000","413683200","783820800","690923520","325140480","87091200","13547520","1198080","55296","1024"],["878169600","5588352000","11975040000","12134707200","6706022400","2171473920","425779200","50688000","3548160","133120","2048"],["11496038400","80791603200","193197312000","221298739200","140507136000","53392711680","12645642240","1895731200","178421760","10137600","315392","4096"],["161902540800","1245404160000","3287866982400","4201163366400","3013878067200","1315146792960","365318553600","66105262080","7828254720","597442560","28114944","737280","8192"],["2440992153600","20399720140800","58932524851200","83109970944000","66487976755200","32769074257920","10448110632960","2216265891840","316609413120","30381711360","1918844928","76038144","1703936","16384"],["39230231040000","353943862272000","1110651429888000","1713576491827200","1512601471180800","831099709440000","299195895398400","72820165017600","12189462405120","1410351022080","111932620800","5948866560","201277440","3899392","32768"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","12","84","688","6480","69024","820064","10745088","153835776","2388037120","39933915648","715413792768","13665240283136","277153623355392","5946837156188160"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["0","1"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["2","3","2"],["4","5","0"],["2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"],["0","2","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184461,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-fbe17c42884360fd"]},{"id":184462,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-fbe17c42884360fd"]},{"id":184463,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-fbe17c42884360fd"]}],"region_exact":{},"incoming_edges":[{"id":"e-fbe17c42884360fd","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-32c953bc446d5813","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":"cca810e9b85097deb304e267162a941559daf6ec33b955e2faa2d04a80563f64","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cbd7c2e66114e926","role":"sequence","name":"Triangle read by rows: T(n,k) = 2^k * (n!/k!)*binomial(n-1,k-1).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        2;\n  1:        8,         4;\n  2:       36,        40,         8;\n  3:      192,       336,       144,        16;\n  4:     1200,      2880,      1920,       448,        32;\n  5:     8640,     26400,     24000,      8640,      1280,       64;\n  6:    70560,    262080,    302400,    147840,     33600,     3456,      128;\n  7:   645120,   2822400,   3951360,   2446080,    752640,   118272,     8960,     256;\n  8:  6531840,  32901120,  54190080,  40642560,  15805440,  3354624,   387072,   22528,   512;\n  9: 72576000, 413683200, 783820800, 690923520, 325140480, 87091200, 13547520, 1198080, 55296, 1024;\n","recurrence_sentence":"The row polynomials satisfy (2*n+x)*P(n,x) = (2*n^2+4*n+2+(5*n+3)*x+2*x^2)*P(n-1,x) + ((2*n+1)*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.","mathematica":"P[0, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 4*n + 2 + (5*n + 3)*x + 2*x^2)*P[n - 1, x] + ((2*n + 1)*x + x^2)*D[P[n - 1, x], x])/(2*n + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + 4*n + 2 + (5*n + 3)*x + 2*x^2)*v[i - 1] + ((2*n + 1)*x + x^2)*deriv(v[i - 1], x))/(2*n + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cbd7c2e66114e926","steps":["D+1"],"edges":["e-fbe17c42884360fd"],"text":"A086915 \u2192 D+1"},"headline":"A086915 \u2192 D+1","chain":null}